Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #184805 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1866 conjugacy classes of subgroups; only the 322 normal subgroups are shown.
This group has 1866 conjugacy classes of subgroups — too many to draw. Only its 322 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) = 128 |
A minimal faithful permutation representation — 4 generators on 44 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,20)(17,22)(18,21)(23,31)(24,33)(25,32)(26,34)(27,35)(28,36)(29,38)(30,37)(39,42)(40,43)(41,44);(6,9)(10,12)(13,15,18,25)(14,19,21,32)(16,26,27,23)(17,29,30,24)(20,34,35,31)(22,38,37,33)(28,39,41,40)(36,42,44,43);(2,4)(5,7)(6,10)(8,11)(9,12)(13,16,17,28,18,27,30,41)(14,20,22,36,21,35,37,44)(15,23,24,39,25,26,29,40)(19,31,33,42,32,34,38,43);(7,11)(10,12)(14,21)(19,32)(20,35)(22,37)(31,34)(33,38)(36,44)(42,43)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,h^b=hi,d^c=de,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289722);