Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2749 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14375 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1009 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)(4,18)(5,23,25,10)(6,12)(7,11)(9,21)(14,40,42,29)(15,35)(17,38,26,31)(19,46,50,32)(20,33,52,47)(22,48)(24,39,27,36)(28,37)(30,43)(34,54)(41,62)(44,58)(45,57)(49,64)(51,59)(53,60,55,61)(56,63);(1,3)(2,8)(4,19)(5,15)(6,26)(7,16)(9,32)(10,35)(11,13)(12,38)(14,22)(17,28)(18,46)(20,51)(21,50)(23,30)(24,45)(25,43)(27,44)(29,34)(31,37)(33,59)(36,57)(39,58)(40,48)(41,53)(42,54)(47,64)(49,52)(55,63)(56,61)(60,62);(1,4)(2,9,11,18)(3,14,16,42)(5,20)(6,22)(7,21)(8,29)(10,33,23,47)(12,34,37,48)(13,40)(15,41,43,63)(17,19,26,50)(24,53)(25,52)(27,55)(28,54)(30,56)(31,32)(35,62)(36,60,39,61)(38,46)(44,51,45,49)(57,64)(58,59);(1,5)(2,10)(3,15)(4,20)(6,27)(7,25)(8,30)(9,33)(11,23)(12,39)(13,35)(14,41)(16,43)(17,45)(18,47)(19,49)(21,52)(22,55)(24,28)(26,44)(29,56)(31,58)(32,59)(34,61)(36,37)(38,57)(40,62)(42,63)(46,64)(48,60)(50,51)(53,54)Polycyclic presentation:
2^9|a^2=h,c^2=ef,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291459);