Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #467692 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16584 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
649 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,31)(3,14,18,34,19,44,17,9)(4,20,23,10,25,37,24,38)(5,26,28,40,30,11,29,41)(15,45,21,42,48,32,47,35)(16,43,50,57,52,60,51,33)(22,39,55,61,56,53,27,36)(46,62,54,63,64,58,49,59);(1,3,7,18,8,19,6,17)(2,9,31,44,13,34,12,14)(4,21,23,48,25,47,24,15)(5,16,28,50,30,52,29,51)(10,35,20,32,38,42,37,45)(11,33,40,60,26,57,41,43)(22,54,55,64,56,49,27,46)(36,59,53,58,61,63,39,62);(1,4,6,23,8,25,7,24)(2,10,12,20,13,38,31,37)(3,15,17,21,19,48,18,47)(5,27,28,56,30,55,29,22)(9,32,14,42,34,45,44,35)(11,39,40,61,26,53,41,36)(16,49,50,64,52,54,51,46)(33,59,60,62,57,63,43,58);(1,5,8,30)(2,11)(3,16,19,52)(4,22,25,56)(6,29,7,28)(9,33)(10,36)(12,41)(13,26)(14,43)(15,46,48,64)(17,51,18,50)(20,53)(21,54,47,49)(23,55,24,27)(31,40)(32,58)(34,57)(35,59)(37,39)(38,61)(42,63)(44,60)(45,62)Polycyclic presentation:
2^9|a^2=g,b^2=efg,c^2=e,d^2=efi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,h^b=hi,d^c=de,g^c=gi,e^d=ei,f^d=fi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 374889);