Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #195963 |
| 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 | |
1274 conjugacy classes of subgroups; only the 330 normal subgroups are shown.
This group has 1274 conjugacy classes of subgroups — too many to draw. Only its 330 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:
(2,3)(5,6)(7,10)(8,9)(11,12)(13,14,18,22,17,21,23,35)(15,19,26,32,25,31,29,36)(16,20,28,34,27,33,37,43)(24,30,39,42,38,41,40,44);(13,15,23,26,17,25,18,29)(14,19,22,36,21,31,35,32)(16,24,37,39,27,38,28,40)(20,30,34,44,33,41,43,42);(1,2,4,3)(6,9)(10,12)(13,16)(14,20)(15,24)(17,27)(18,28)(19,30)(21,33)(22,34)(23,37)(25,38)(26,39)(29,40)(31,41)(32,42)(35,43)(36,44);(1,3,4,2)(5,7,8,11)(6,10,9,12)(14,21)(16,27)(19,31)(22,35)(24,38)(28,37)(32,36)(39,40)Polycyclic presentation:
2^9|a^2=h,b^2=ehi,c^2=gi,d^2=f,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,e^b=ei,h^b=hi,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300820);