Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512832 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
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 | |
451 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 451 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 132 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 132 points:
(2,4)(5,6,12,18)(7,13,20,42)(8,24,28,77)(9,29,34,75)(10,35,38,91)(11,39,37,17)(14,46,49,25)(15,50,53,32)(16,54,56,31)(19,58,59,111)(21,66,63,110)(22,71,64,119)(23,73,65,45)(26,79,80,108)(27,81,40,76)(30,55,86,109)(33,89,88,83)(36,51,92,107)(41,93,94,60)(43,99,97,69)(44,102,98,68)(47,90,105,85)(48,78,57,84)(52,87,82,106)(61,115,113,132)(62,116,74,112)(67,103,117,130)(70,126,118,122)(72,100,120,129)(95,127,128,124)(96,114,104,123)(101,125,121,131);(1,2,3,4)(5,7)(6,13)(8,19)(9,21)(10,22)(11,23)(12,20)(14,41)(15,43)(16,44)(17,45)(18,42)(24,58)(25,60)(26,61)(27,62)(28,59)(29,66)(30,67)(31,68)(32,69)(33,70)(34,63)(35,71)(36,72)(37,65)(38,64)(39,73)(40,74)(46,93)(47,95)(48,96)(49,94)(50,99)(51,100)(52,101)(53,97)(54,102)(55,103)(56,98)(57,104)(75,110)(76,112)(77,111)(78,114)(79,115)(80,113)(81,116)(82,121)(83,122)(84,123)(85,124)(86,117)(87,125)(88,118)(89,126)(90,127)(91,119)(92,120)(105,128)(106,131)(107,129)(108,132)(109,130);(5,8,12,28)(6,14,18,49)(7,19,20,59)(9,30,34,86)(10,36,38,92)(11,40,37,27)(13,41,42,94)(15,51,53,107)(16,55,56,109)(17,57,39,48)(21,67,63,117)(22,72,64,120)(23,74,65,62)(24,75,77,29)(25,50,46,32)(26,54,80,31)(33,90,88,85)(35,47,91,105)(43,100,97,129)(44,103,98,130)(45,104,73,96)(52,108,82,79)(58,110,111,66)(60,99,93,69)(61,102,113,68)(70,127,118,124)(71,95,119,128)(76,83,81,89)(78,87,84,106)(101,132,121,115)(112,122,116,126)(114,125,123,131);(7,20)(13,42)(19,59)(21,63)(22,64)(23,65)(41,94)(43,97)(44,98)(45,73)(58,111)(60,93)(61,113)(62,74)(66,110)(67,117)(68,102)(69,99)(70,118)(71,119)(72,120)(95,128)(96,104)(100,129)(101,121)(103,130)(112,116)(114,123)(115,132)(122,126)(124,127)(125,131)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,f^a=fg,g^a=gh,h^a=hi,d^b=di,f^c=fg,g^c=gh,h^c=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420118);