Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #184767 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×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 | |
2480 conjugacy classes of subgroups; only the 378 normal subgroups are shown.
This group has 2480 conjugacy classes of subgroups — too many to draw. Only its 378 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)(13,14,17,22,18,21,30,35)(15,23,25,36,26,19,29,32)(16,20,28,34,27,33,40,43)(24,37,39,44,38,31,41,42);(5,6)(7,10)(8,9)(11,12)(13,15,18,26)(14,19,21,23)(16,24,27,38)(17,29,30,25)(20,31,33,37)(22,36,35,32)(28,41,40,39)(34,44,43,42);(2,4)(6,9)(10,12)(13,16,18,27)(14,20,21,33)(15,24,26,38)(17,28,30,40)(19,31,23,37)(22,34,35,43)(25,39,29,41)(32,42,36,44);(5,7,8,11)(6,10,9,12)(16,27)(20,33)(24,38)(28,40)(31,37)(34,43)(39,41)(42,44)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=i,d^2=g,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,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, 266046);