Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499897 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
703 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 703 conjugacy classes of subgroups — too many to draw. Only its 152 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| 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)(15,23)(16,26)(17,28)(19,31)(21,36)(22,35)(24,41)(25,40)(27,30)(29,39)(32,43)(33,44)(34,42)(37,38);(1,2,3,4)(6,9)(10,11)(13,14)(15,19)(16,21)(17,22)(18,20)(23,31)(24,33)(25,32)(26,36)(27,37)(28,35)(29,34)(30,38)(39,42)(40,43)(41,44);(2,4)(13,15,18,25)(14,19,20,32)(16,27,29,41)(17,30,28,24)(21,37,34,44)(22,38,35,33)(23,39,40,26)(31,42,43,36);(5,7,8,12)(6,10,9,11)(14,20)(19,32)(21,34)(22,35)(31,43)(33,38)(36,42)(37,44)Polycyclic presentation:
2^9|b^2=g,c^2=i,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,d^b=di,f^c=fh,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 413481);