Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #513098 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D16 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| prime radical | C2×#508 | C22 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
380 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 136 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 136 points:
(1,2)(3,7)(4,6)(5,8)(9,10,16,22)(11,17,24,46)(12,28,32,81)(13,33,38,79)(14,39,42,95)(15,43,41,21)(18,50,53,29)(19,54,57,36)(20,58,60,35)(23,62,63,115)(25,70,67,114)(26,75,68,123)(27,77,69,49)(30,83,84,112)(31,85,44,80)(34,59,90,113)(37,93,92,87)(40,55,96,111)(45,97,98,64)(47,103,101,73)(48,106,102,72)(51,94,109,89)(52,82,61,88)(56,91,86,110)(65,119,117,136)(66,120,78,116)(71,107,121,134)(74,130,122,126)(76,104,124,133)(99,131,132,128)(100,118,108,127)(105,129,125,135);(1,3,4,8)(2,5,6,7)(9,11)(10,17)(12,23)(13,25)(14,26)(15,27)(16,24)(18,45)(19,47)(20,48)(21,49)(22,46)(28,62)(29,64)(30,65)(31,66)(32,63)(33,70)(34,71)(35,72)(36,73)(37,74)(38,67)(39,75)(40,76)(41,69)(42,68)(43,77)(44,78)(50,97)(51,99)(52,100)(53,98)(54,103)(55,104)(56,105)(57,101)(58,106)(59,107)(60,102)(61,108)(79,114)(80,116)(81,115)(82,118)(83,119)(84,117)(85,120)(86,125)(87,126)(88,127)(89,128)(90,121)(91,129)(92,122)(93,130)(94,131)(95,123)(96,124)(109,132)(110,135)(111,133)(112,136)(113,134);(3,8)(5,7)(9,12,16,32)(10,18,22,53)(11,23,24,63)(13,34,38,90)(14,40,42,96)(15,44,41,31)(17,45,46,98)(19,55,57,111)(20,59,60,113)(21,61,43,52)(25,71,67,121)(26,76,68,124)(27,78,69,66)(28,79,81,33)(29,54,50,36)(30,58,84,35)(37,94,92,89)(39,51,95,109)(47,104,101,133)(48,107,102,134)(49,108,77,100)(56,112,86,83)(62,114,115,70)(64,103,97,73)(65,106,117,72)(74,131,122,128)(75,99,123,132)(80,87,85,93)(82,91,88,110)(105,136,125,119)(116,126,120,130)(118,129,127,135);(2,6)(5,7)(11,24)(17,46)(23,63)(25,67)(26,68)(27,69)(45,98)(47,101)(48,102)(49,77)(62,115)(64,97)(65,117)(66,78)(70,114)(71,121)(72,106)(73,103)(74,122)(75,123)(76,124)(99,132)(100,108)(104,133)(105,125)(107,134)(116,120)(118,127)(119,136)(126,130)(128,131)(129,135)Polycyclic presentation:
2^9|a^2=i,b^2=e,c^2=i,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fg,g^a=gh,h^a=hi,c^b=ce,d^b=di,f^c=fg,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420319);