Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #439368 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15635 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,14,7,12,8,13)(3,9,29,33,17,34,18,39)(4,19,28,36,22,38,23,10)(5,24,26,11,27,40,30,37)(15,41,43,31,20,49,44,57)(16,45,47,32,25,54,46,58)(21,35,53,48,51,60,52,55)(42,59,62,61,50,56,63,64);(1,3,8,29,7,17,6,18)(2,9,14,39,12,34,13,33)(4,20,28,43,22,15,23,44)(5,25,26,47,27,16,30,46)(10,31,38,49,36,57,19,41)(11,32,24,54,37,58,40,45)(21,50,52,63,51,42,53,62)(35,59,48,64,60,56,55,61);(1,4,8,23,7,22,6,28)(2,10,13,38,12,36,14,19)(3,15,18,44,17,20,29,43)(5,21,30,52,27,51,26,53)(9,31,39,41,34,57,33,49)(11,35,24,55,37,60,40,48)(16,42,46,63,25,50,47,62)(32,56,45,61,58,59,54,64);(1,5)(2,11)(3,16)(4,21)(6,26)(7,27)(8,30)(9,32)(10,35)(12,37)(13,24)(14,40)(15,42)(17,25)(18,46)(19,48)(20,50)(22,51)(23,52)(28,53)(29,47)(31,56)(33,54)(34,58)(36,60)(38,55)(39,45)(41,61)(43,62)(44,63)(49,64)(57,59)Polycyclic presentation:
2^9|a^2=ef,b^2=efg,c^2=efi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 346565);