Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30871 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2675 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14521 |
| prime radical | #23438 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1198 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,14,6,13)(3,15,18,9,20,33,19,34)(4,21,25,10,26,37,24,38)(5,11,29,40,30,42,28,41)(16,43,45,35,46,31,22,51)(17,47,27,57,48,39,49,32)(23,53,55,62,56,50,54,36)(44,61,58,60,63,59,52,64);(1,3)(2,9)(4,22)(5,27)(6,19)(7,18)(8,20)(10,35)(11,39)(12,33)(13,15)(14,34)(16,25)(17,28)(21,43)(23,44)(24,46)(26,45)(29,48)(30,49)(31,37)(32,40)(36,61)(38,51)(41,57)(42,47)(50,64)(52,54)(53,60)(55,58)(56,63)(59,62);(1,4,7,25,8,26,6,24)(2,10,12,37,14,38,13,21)(3,16,18,45,20,46,19,22)(5,23,29,55,30,56,28,54)(9,31,33,51,34,43,15,35)(11,36,40,53,42,62,41,50)(17,44,27,58,48,63,49,52)(32,59,47,64,57,61,39,60);(1,5,6,28,8,30,7,29)(2,11,12,40,14,42,13,41)(3,17,18,27,20,48,19,49)(4,23,24,54,26,56,25,55)(9,32,15,39,34,57,33,47)(10,36,37,53,38,62,21,50)(16,44,45,58,46,63,22,52)(31,59,35,60,43,61,51,64)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,f^c=fi,g^c=gi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295682);