Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C4⋊C4 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1192 conjugacy classes of subgroups; only the 338 normal subgroups are shown.
This group has 1192 conjugacy classes of subgroups — too many to draw. Only its 338 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)(13,14,18,22,19,23,17,32)(15,24,28,34,26,38,27,20)(16,21,31,36,29,35,30,42)(25,37,40,43,41,44,39,33);(5,6)(7,10)(8,9)(11,12)(13,15,17,27,19,26,18,28)(14,20,22,24,23,34,32,38)(16,25,30,39,29,41,31,40)(21,33,36,37,35,43,42,44);(2,4)(6,9)(10,12)(13,16)(14,21)(15,25)(17,30)(18,31)(19,29)(20,33)(22,36)(23,35)(24,37)(26,41)(27,39)(28,40)(32,42)(34,43)(38,44);(5,7,8,11)(6,10,9,12)(15,26)(16,29)(20,34)(21,35)(24,38)(27,28)(30,31)(36,42)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=g,e^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,d^b=di,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266092);