Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501046 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#466 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
659 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 44 points:
(5,6)(7,11)(8,9)(10,12)(13,14,16,21,28,35,41,39,18,20,30,34,17,22,29,37)(15,23,24,38,31,44,43,32,26,40,42,33,25,36,27,19);(1,2)(3,4)(6,9)(10,11);(2,4)(13,15,18,26)(14,19,20,32)(16,27,30,43)(17,31,28,25)(21,36,34,44)(22,38,35,33)(23,39,40,37)(24,41,42,29);(5,7,8,12)(6,10,9,11)(14,20)(19,32)(21,34)(22,35)(23,40)(33,38)(36,44)(37,39)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fh,h^c=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408701);