Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31282 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2684 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14525 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
787 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,30)(3,14,19,34,18,44,17,9)(4,20,24,10,25,37,23,38)(5,26,28,40,29,11,31,41)(15,32,47,42,21,35,46,59)(16,39,27,33,48,43,49,60)(22,50,53,36,55,63,54,56)(45,58,57,62,51,61,52,64);(1,3,6,17,8,18,7,19)(2,9,12,14,13,34,30,44)(4,21,24,46,25,15,23,47)(5,27,28,48,29,49,31,16)(10,35,20,42,38,32,37,59)(11,39,40,60,26,43,41,33)(22,52,54,51,55,57,53,45)(36,62,63,61,56,64,50,58);(1,4,7,24,8,25,6,23)(2,10,12,37,13,38,30,20)(3,15,19,47,18,21,17,46)(5,22,28,53,29,55,31,54)(9,32,14,42,34,35,44,59)(11,36,41,63,26,56,40,50)(16,45,27,57,48,51,49,52)(33,58,43,62,60,61,39,64);(1,5)(2,11)(3,16)(4,22)(6,31)(7,28)(8,29)(9,33)(10,36)(12,41)(13,26)(14,43)(15,45)(17,49)(18,48)(19,27)(20,50)(21,51)(23,54)(24,53)(25,55)(30,40)(32,58)(34,60)(35,61)(37,63)(38,56)(39,44)(42,62)(46,52)(47,57)(59,64)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=h,e^2=i,f^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,g^b=gi,f^c=fi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295899);