Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #403153 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14868 |
| 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 | |
1164 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,6,13,7,26)(3,14,18,32,17,9,27,38)(4,10,28,35,21,36,22,39)(5,23,29,52,24,11,25,37)(15,30,19,47,43,40,46,33)(16,31,45,59,44,41,53,56)(20,49,54,60,50,34,51,58)(42,61,48,57,62,55,63,64);(1,3,6,17)(2,9)(4,19,21,46)(5,16,24,44)(7,27,8,18)(10,33)(11,31)(12,38)(13,14)(15,28,43,22)(20,48,50,63)(23,41)(25,53,29,45)(26,32)(30,35)(34,57)(36,47)(37,59)(39,40)(42,54,62,51)(49,64)(52,56)(55,58)(60,61);(1,4,6,21)(2,10,13,36)(3,15,17,43)(5,20,24,50)(7,28,8,22)(9,30,14,40)(11,34,23,49)(12,39,26,35)(16,42,44,62)(18,46,27,19)(25,54,29,51)(31,55,41,61)(32,47,38,33)(37,60,52,58)(45,63,53,48)(56,64,59,57);(1,5,6,24)(2,11)(3,16,17,44)(4,20,21,50)(7,25,8,29)(9,31)(10,34)(12,37)(13,23)(14,41)(15,42,43,62)(18,45,27,53)(19,48,46,63)(22,51,28,54)(26,52)(30,55)(32,56)(33,57)(35,58)(36,49)(38,59)(39,60)(40,61)(47,64)Polycyclic presentation:
2^9|a^2=fg,b^2=ef,c^2=i,d^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 310350);