Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396106 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2459 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14745 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
948 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)(3,14)(4,10)(5,24)(6,28)(7,13)(8,12)(9,17)(11,26)(15,39)(16,44)(18,41)(19,32)(20,30)(21,50)(22,36)(23,37)(25,38)(27,54)(29,35)(31,47)(33,43)(34,52)(40,46)(42,58)(45,53)(48,57)(49,64)(51,59)(55,60)(56,63)(61,62);(1,3,8,19,7,18,6,17)(2,9,28,41,13,32,12,14)(4,20,22,43,23,48,29,15)(5,16,27,47,26,46,25,45)(10,33,35,30,37,39,36,57)(11,31,54,44,24,53,38,40)(21,49,51,62,52,63,55,42)(34,58,59,56,50,61,60,64);(1,4,6,22,7,23,8,29)(2,10,12,35,13,37,28,36)(3,15,17,20,18,43,19,48)(5,21,25,51,26,52,27,55)(9,30,14,39,32,57,41,33)(11,34,38,59,24,50,54,60)(16,42,45,49,46,62,47,63)(31,56,40,61,53,64,44,58);(1,5,6,25,7,26,8,27)(2,11,12,38,13,24,28,54)(3,16,17,45,18,46,19,47)(4,21,22,51,23,52,29,55)(9,31,14,40,32,53,41,44)(10,34,35,59,37,50,36,60)(15,42,20,49,43,62,48,63)(30,56,39,61,57,64,33,58)Polycyclic presentation:
2^9|b^2=ef,c^2=e,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^c=fi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 304306);