Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #360200 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2311 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13872 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1533 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,28)(3,14,19,32,18,40,17,9)(4,20,22,35,23,10,29,36)(5,24,26,37,27,54,25,11)(15,38,42,30,43,57,48,33)(16,44,47,60,46,31,45,39)(21,50,51,49,52,53,55,34)(41,58,61,59,62,56,63,64);(1,3,6,17,8,18,7,19)(2,9,12,14,13,32,28,40)(4,15,22,42,23,43,29,48)(5,16,25,45,27,46,26,47)(10,33,35,57,20,30,36,38)(11,31,24,39,37,44,54,60)(21,41,51,61,52,62,55,63)(34,58,53,64,49,56,50,59);(1,4)(2,10)(3,15)(5,21)(6,22)(7,29)(8,23)(9,30)(11,34)(12,35)(13,20)(14,38)(16,41)(17,42)(18,43)(19,48)(24,53)(25,51)(26,55)(27,52)(28,36)(31,56)(32,33)(37,49)(39,59)(40,57)(44,58)(45,61)(46,62)(47,63)(50,54)(60,64);(1,5,6,25,8,27,7,26)(2,11,12,24,13,37,28,54)(3,16,17,45,18,46,19,47)(4,21,22,51,23,52,29,55)(9,31,14,39,32,44,40,60)(10,34,35,53,20,49,36,50)(15,41,42,61,43,62,48,63)(30,56,38,59,33,58,57,64)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267397);