Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458026 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16308 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
891 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14)(3,15,18,9)(4,21,25,38)(5,27,29,40)(6,13,7,12)(10,26,37,24)(11,30,39,28)(16,42,22,52)(17,46,48,57)(19,33,20,34)(23,53,55,36)(31,44,35,45)(32,49,41,50)(43,58,47,59)(51,56,60,54)(61,63,64,62);(1,3,6,19,8,18,7,20)(2,9,12,33,14,15,13,34)(4,22,24,45,25,16,26,44)(5,17,28,49,29,48,30,50)(10,35,21,42,37,31,38,52)(11,32,27,57,39,41,40,46)(23,47,54,63,55,43,56,62)(36,59,51,61,53,58,60,64);(1,4,6,24,8,25,7,26)(2,10,12,21,14,37,13,38)(3,16,20,45,18,22,19,44)(5,23,28,54,29,55,30,56)(9,31,34,42,15,35,33,52)(11,36,27,51,39,53,40,60)(17,47,50,62,48,43,49,63)(32,59,46,64,41,58,57,61);(1,5)(2,11)(3,17)(4,23)(6,28)(7,30)(8,29)(9,32)(10,36)(12,27)(13,40)(14,39)(15,41)(16,43)(18,48)(19,49)(20,50)(21,51)(22,47)(24,54)(25,55)(26,56)(31,58)(33,57)(34,46)(35,59)(37,53)(38,60)(42,61)(44,62)(45,63)(52,64)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,e^b=ei,d^c=de,g^c=gi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 365223);