Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499897 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
545 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 545 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) = 512 |
A minimal faithful permutation representation — 4 generators on 44 points:
(5,6,8,9)(7,11,12,10)(13,14,16,20)(15,23,24,42)(17,29,27,41)(18,32,28,22)(19,34,35,25)(21,38,37,31)(26,43,33,39)(30,40,44,36);(1,2,3,4)(6,9)(10,11)(13,14)(15,19)(16,20)(17,21)(18,22)(23,25)(24,35)(26,36)(27,37)(28,32)(29,31)(30,39)(33,40)(34,42)(38,41)(43,44);(2,4)(13,15,16,24)(14,19,20,35)(17,30,27,44)(18,33,28,26)(21,39,37,43)(22,40,32,36)(23,41,42,29)(25,38,34,31);(5,7,8,12)(6,10,9,11)(14,20)(19,35)(21,37)(22,32)(23,42)(29,41)(36,40)(39,43)Polycyclic presentation:
2^9|a^2=e,b^2=g,c^2=i,d^2=ei,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, 413543);