Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #184805 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×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 | C23×#41 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3482 conjugacy classes of subgroups; only the 362 normal subgroups are shown.
This group has 3482 conjugacy classes of subgroups — too many to draw. Only its 362 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)(16,26)(17,30)(18,21)(20,28)(22,32)(23,36)(24,38)(25,34)(27,33)(29,37)(31,35)(39,43)(40,42)(41,44);(6,9)(10,12)(13,15,18,25)(14,19,21,34)(16,27,28,23)(17,31,32,24)(20,36,26,33)(22,38,30,35)(29,39,42,41)(37,43,40,44);(2,4)(5,7)(6,10)(8,11)(9,12)(13,16,17,29,18,28,32,42)(14,20,22,37,21,26,30,40)(15,23,24,39,25,27,31,41)(19,33,35,43,34,36,38,44);(7,11)(10,12)(14,21)(19,34)(20,26)(22,30)(33,36)(35,38)(37,40)(43,44)Polycyclic presentation:
2^9|b^2=i,c^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,h^a=hi,c^b=cg,h^b=hi,d^c=de,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289595);