Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #380127 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14259 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1103 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,7,11,25,34)(3,13,17,38,16,8,26,29)(4,18,20,35,21,9,24,31)(5,10,23,33,22,32,49,58)(14,27,41,53,40,36,45,56)(15,37,44,60,43,28,51,55)(19,46,48,59,47,30,50,57)(39,52,61,63,42,54,62,64);(1,3)(2,8,11,13)(4,14)(5,15)(6,17)(7,16)(9,27,18,36)(10,28,32,37)(12,29,34,38)(19,39)(20,41)(21,40)(22,43)(23,44)(24,45)(25,26)(30,52,46,54)(31,53,35,56)(33,55,58,60)(42,47)(48,61)(49,51)(50,62)(57,63,59,64);(1,4,7,21)(2,9,11,18)(3,14,16,40)(5,19,22,47)(6,24,25,20)(8,27,13,36)(10,30,32,46)(12,35,34,31)(15,42,43,39)(17,45,26,41)(23,50,49,48)(28,54,37,52)(29,56,38,53)(33,59,58,57)(44,61,51,62)(55,63,60,64);(1,5)(2,10)(3,15)(4,19)(6,23)(7,22)(8,28)(9,30)(11,32)(12,33)(13,37)(14,39)(16,43)(17,44)(18,46)(20,48)(21,47)(24,50)(25,49)(26,51)(27,52)(29,55)(31,57)(34,58)(35,59)(36,54)(38,60)(40,42)(41,61)(45,62)(53,63)(56,64)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,e^b=ei,d^c=de,h^c=hi,g^d=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287325);