Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153083 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2215 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| prime radical | C2×#41 | D8 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
345 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 345 conjugacy classes of subgroups — too many to draw. Only its 72 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 — 3 generators on 132 points:
(1,2)(3,4)(5,6,12,18)(7,19,24,65)(8,25,31,52)(9,32,37,62)(10,38,40,16)(11,41,43,99)(13,44,48,21)(14,30,53,28)(15,54,57,36)(17,59,61,34)(20,66,70,124)(22,72,39,64)(23,73,74,119)(26,63,69,125)(27,79,85,123)(29,51,87,114)(33,60,94,120)(35,96,98,89)(42,55,100,118)(45,104,107,77)(46,71,58,93)(47,97,108,92)(49,103,106,67)(50,110,113,130)(56,95,90,116)(68,117,126,132)(75,115,112,83)(76,86,101,81)(78,127,102,111)(80,121,129,109)(82,88,84,122)(91,105,128,131);(5,7,12,24)(6,13,18,48)(8,26,31,69)(9,33,37,94)(10,39,40,22)(11,42,43,100)(14,49,53,106)(15,55,57,118)(16,58,38,46)(17,60,61,120)(19,62,65,32)(20,30,70,28)(21,54,44,36)(23,59,74,34)(25,45,52,107)(27,80,85,129)(29,68,87,126)(35,97,98,92)(41,47,99,108)(50,111,113,127)(51,105,114,131)(56,119,90,73)(63,123,125,79)(64,89,72,96)(66,122,124,88)(67,110,103,130)(71,95,93,116)(75,117,112,132)(76,121,101,109)(77,115,104,83)(78,86,102,81)(82,91,84,128);(2,4)(5,8)(6,14)(7,20)(9,27)(10,30)(11,29)(12,31)(13,45)(15,50)(16,52)(17,51)(18,53)(19,63)(21,67)(22,69)(23,68)(24,70)(25,38)(26,39)(28,40)(32,88)(33,91)(34,81)(35,84)(36,83)(37,85)(41,101)(42,102)(43,87)(44,103)(46,106)(47,105)(48,107)(49,58)(54,115)(55,117)(56,112)(57,113)(59,86)(60,121)(61,114)(62,122)(64,124)(65,125)(66,72)(71,104)(73,127)(74,126)(75,90)(76,99)(77,93)(78,100)(79,96)(80,97)(82,98)(89,123)(92,129)(94,128)(95,110)(108,131)(109,120)(111,119)(116,130)(118,132)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=gh,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fi,g^a=gh,h^a=hi,c^b=cf,d^b=dg,e^b=ei,f^b=fi,g^b=gh,h^b=hi,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60281);