Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #401802 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14823 |
| 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 | |
1129 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,13,8,12,7,30)(3,14,17,34,18,9,29,33)(4,19,22,39,24,10,23,37)(5,25,27,38,26,56,28,11)(15,31,20,35,43,40,47,48)(16,44,46,41,45,55,57,32)(21,51,53,36,52,60,54,49)(42,62,50,58,63,59,64,61);(1,3)(2,9)(4,20)(5,16)(6,29)(7,17)(8,18)(10,35)(11,32)(12,14)(13,34)(15,22)(19,48)(21,50)(23,43)(24,47)(25,55)(26,45)(27,57)(28,46)(30,33)(31,37)(36,59)(38,41)(39,40)(42,53)(44,56)(49,62)(51,61)(52,64)(54,63)(58,60);(1,4,8,24)(2,10,12,19)(3,15,18,43)(5,21,26,52)(6,23,7,22)(9,31,14,40)(11,36,38,49)(13,39,30,37)(16,42,45,63)(17,47,29,20)(25,51,56,60)(27,54,28,53)(32,58,41,61)(33,48,34,35)(44,62,55,59)(46,64,57,50);(1,5)(2,11,12,38)(3,16)(4,21)(6,27)(7,28)(8,26)(9,32,14,41)(10,36,19,49)(13,25,30,56)(15,42)(17,46)(18,45)(20,50)(22,53)(23,54)(24,52)(29,57)(31,58,40,61)(33,44,34,55)(35,59,48,62)(37,60,39,51)(43,63)(47,64)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,g^b=gi,g^c=gi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308999);