Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511747 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | C2×D8 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
623 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 44 points:
(2,3)(5,6)(7,9)(8,10)(11,12)(13,14,18,23)(15,19,24,31)(16,20,29,36)(17,30,28,22)(21,37,35,27)(25,32,38,42)(26,41,39,34)(33,44,43,40);(1,2,4,3)(5,7,8,11)(6,9,10,12)(15,24)(19,31)(25,38)(26,39)(32,42)(33,43)(34,41)(40,44);(6,10)(9,12)(13,15,18,24)(14,19,23,31)(16,25,29,38)(17,26,28,39)(20,32,36,42)(21,33,35,43)(22,34,30,41)(27,40,37,44);(1,3,4,2)(14,20)(16,27)(17,28)(19,32)(21,22)(23,36)(25,40)(26,39)(29,37)(30,35)(31,42)(33,34)(38,44)(41,43)Polycyclic presentation:
2^9|a^2=i,b^2=ef,c^2=i,d^2=e,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gh,h^a=hi,c^b=ci,g^d=gh,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419486);