Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #434065 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15496 |
| 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 | |
1126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,29,6,12,7,13)(3,14,30,34,17,9,18,42)(4,19,22,49,23,10,31,37)(5,24,28,39,26,11,27,40)(15,41,20,47,44,32,51,35)(16,33,46,58,25,38,45,57)(21,36,52,60,53,48,54,61)(43,56,63,64,55,62,50,59);(1,3,6,17)(2,9,12,14)(4,20)(5,25,26,16)(7,18,8,30)(10,35)(11,38,24,33)(13,34,29,42)(15,31)(19,47)(21,50)(22,44)(23,51)(27,46,28,45)(32,49)(36,59)(37,41)(39,58,40,57)(43,52)(48,64)(53,63)(54,55)(56,60)(61,62);(1,4)(2,10,12,19)(3,15,17,44)(5,21)(6,23)(7,22)(8,31)(9,32)(11,36,24,48)(13,37,29,49)(14,41)(16,43,25,55)(18,20,30,51)(26,53)(27,52)(28,54)(33,56)(34,35)(38,62)(39,60,40,61)(42,47)(45,63,46,50)(57,64)(58,59);(1,5,6,26)(2,11,12,24)(3,16,17,25)(4,21,23,53)(7,28,8,27)(9,33,14,38)(10,36,19,48)(13,40,29,39)(15,43,44,55)(18,46,30,45)(20,50,51,63)(22,54,31,52)(32,56,41,62)(34,58,42,57)(35,59,47,64)(37,61,49,60)Polycyclic presentation:
2^9|a^2=eg,b^2=e,c^2=ef,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^b=fi,e^c=ei,f^c=fi,g^d=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341262);