Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384166 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2377 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14417 |
| prime radical | #60453 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1385 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,13,16,8)(4,9)(5,22,24,10)(6,25)(7,11)(12,27)(14,28,40,38)(15,35)(17,30,26,37)(18,43,45,31)(19,46,48,32)(20,49)(21,33)(23,36,53,51)(29,41)(34,50)(39,54)(42,55)(44,57)(47,58,64,63)(52,60)(56,62)(59,61);(1,3)(2,8)(4,18)(5,15)(6,26)(7,16)(9,31)(10,35)(11,13)(12,37)(14,20)(17,27)(19,44)(21,45)(22,29)(23,52)(24,41)(25,30)(28,34)(32,57)(33,43)(36,55)(38,49)(39,47)(40,50)(42,53)(46,56)(48,62)(51,60)(54,63)(58,59)(61,64);(1,4,6,20,7,21,27,50)(2,9,25,49,11,33,12,34)(3,14,17,18,16,40,26,45)(5,19,23,47,24,48,53,64)(8,28,37,43,13,38,30,31)(10,32,51,63,22,46,36,58)(15,39,42,44,41,61,52,62)(29,54,60,57,35,59,55,56);(1,5)(2,10)(3,15)(4,19)(6,23)(7,24)(8,29)(9,32)(11,22)(12,36)(13,35)(14,39)(16,41)(17,42)(18,44)(20,47)(21,48)(25,51)(26,52)(27,53)(28,54)(30,55)(31,56)(33,46)(34,58)(37,60)(38,59)(40,61)(43,57)(45,62)(49,63)(50,64)Polycyclic presentation:
2^9|a^2=h,c^2=eg,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293062);