Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #360226 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2352 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13872 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1523 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,14)(3,15,19,33,20,40,18,9)(4,21,24,36,26,10,25,37)(5,27,29,38,30,39,28,11)(16,34,22,51,44,41,49,31)(17,45,47,60,48,32,46,42)(23,53,54,50,56,57,55,35)(43,58,52,59,62,63,64,61);(1,3,6,18,8,20,7,19)(2,9,12,15,13,33,14,40)(4,22,24,44,26,49,25,16)(5,17,28,46,30,48,29,47)(10,34,36,31,21,41,37,51)(11,32,27,42,38,45,39,60)(23,52,54,62,56,64,55,43)(35,59,57,58,50,61,53,63);(1,4)(2,10)(3,16)(5,23)(6,24)(7,25)(8,26)(9,31)(11,35)(12,36)(13,21)(14,37)(15,41)(17,43)(18,22)(19,49)(20,44)(27,57)(28,54)(29,55)(30,56)(32,58)(33,51)(34,40)(38,50)(39,53)(42,61)(45,63)(46,52)(47,64)(48,62)(59,60);(1,5,6,28,8,30,7,29)(2,11,12,27,13,38,14,39)(3,17,18,46,20,48,19,47)(4,23,24,54,26,56,25,55)(9,32,15,42,33,45,40,60)(10,35,36,57,21,50,37,53)(16,43,22,52,44,62,49,64)(31,58,41,61,51,63,34,59)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267423);