Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #184786 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
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 | |
3366 conjugacy classes of subgroups; only the 442 normal subgroups are shown.
This group has 3366 conjugacy classes of subgroups — too many to draw. Only its 442 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)(13,14,17,22,18,20,30,35)(15,19,24,34,25,32,40,42)(16,26,28,38,29,21,31,37)(23,36,39,44,27,33,41,43);(6,9)(10,12)(13,15,18,25)(14,19,20,32)(16,27,29,23)(17,24,30,40)(21,36,26,33)(22,34,35,42)(28,41,31,39)(37,44,38,43);(2,4)(5,7,8,11)(6,10,9,12)(14,20)(17,30)(19,32)(21,26)(24,40)(28,31)(33,36)(39,41);(13,16,18,29)(14,21,20,26)(15,23,25,27)(17,31,30,28)(19,33,32,36)(22,38,35,37)(24,41,40,39)(34,44,42,43)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=e,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,g^a=gi,d^b=di,h^c=hi,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299450);