Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #195981 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator, prime radical | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1194 conjugacy classes of subgroups; only the 298 normal subgroups are shown.
This group has 1194 conjugacy classes of subgroups — too many to draw. Only its 298 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:
(6,9)(10,12)(13,14,17,22,18,21,30,35)(15,19,25,33,24,32,40,43)(16,26,27,36,28,20,29,34)(23,37,39,44,38,31,41,42);(1,2,3,4)(5,6,8,9)(7,10,11,12)(13,15)(14,19)(16,23)(17,25)(18,24)(20,31)(21,32)(22,33)(26,37)(27,39)(28,38)(29,41)(30,40)(34,42)(35,43)(36,44);(2,4)(7,11)(10,12)(13,16,18,28)(14,20,21,26)(15,23,24,38)(17,29,30,27)(19,31,32,37)(22,36,35,34)(25,41,40,39)(33,44,43,42);(5,7,8,11)(6,10,9,12)(15,24)(19,32)(23,38)(25,40)(31,37)(33,43)(39,41)(42,44)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=i,d^2=e,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,d^b=di,d^c=de,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287878);