Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153086 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2184 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| prime radical | C2×#508 | C22 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
361 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 361 conjugacy classes of subgroups — too many to draw. Only its 76 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:
(5,6,12,18)(7,19,24,71)(8,25,31,53)(9,32,38,68)(10,39,42,16)(11,43,46,110)(13,47,51,21)(14,27,56,28)(15,57,61,37)(17,64,67,35)(20,48,75,116)(22,78,40,70)(23,80,82,127)(26,86,79,52)(29,95,41,85)(30,96,98,128)(33,65,103,129)(34,66,92,84)(36,106,109,99)(44,58,111,126)(45,59,90,122)(49,77,62,102)(50,107,118,101)(54,94,63,91)(55,108,124,104)(60,105,100,125)(69,72,130,73)(74,117,97,132)(76,114,112,113)(81,115,131,123)(83,88,121,87)(89,120,93,119);(1,2)(3,4)(5,7,12,24)(6,13,18,51)(8,26,31,79)(9,33,38,103)(10,40,42,22)(11,44,46,111)(14,52,56,86)(15,58,61,126)(16,62,39,49)(17,65,67,129)(19,68,71,32)(20,41,75,29)(21,57,47,37)(23,64,82,35)(25,83,53,121)(27,89,28,93)(30,97,98,74)(34,88,92,87)(36,107,109,101)(43,50,110,118)(45,81,90,131)(48,63,116,54)(55,123,124,115)(59,120,122,119)(60,127,100,80)(66,117,84,132)(69,95,130,85)(70,99,78,106)(72,104,73,108)(76,91,112,94)(77,105,102,125)(96,114,128,113);(2,4)(5,8)(6,14)(7,20)(9,34)(10,41)(11,45)(12,31)(13,48)(15,59)(16,63)(17,66)(18,56)(19,69)(21,76)(22,79)(23,81)(24,75)(25,32)(26,40)(27,57)(28,37)(29,42)(30,35)(33,73)(36,108)(38,92)(39,54)(43,55)(44,74)(46,90)(47,112)(49,86)(50,117)(51,116)(52,62)(53,68)(58,114)(60,128)(61,122)(64,98)(65,115)(67,84)(70,121)(71,130)(72,103)(77,89)(78,83)(80,119)(82,131)(85,106)(87,101)(88,107)(91,105)(93,102)(94,125)(95,99)(96,100)(97,111)(104,109)(110,124)(113,126)(118,132)(120,127)(123,129)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=gh,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fi,g^a=gh,h^a=hi,c^b=cf,d^b=dg,e^b=eg,f^b=fi,g^b=gh,h^b=hi,d^c=dgi,e^c=eg,f^c=fi,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, 60312);