Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C23 | C2×C22⋊C8 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
556 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 556 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 256 |
A minimal faithful permutation representation — 3 generators on 44 points:
(1,2,4,3)(5,6)(7,11)(8,10)(9,12)(13,14,17,24,19,26,20,23)(15,27,30,38,32,39,29,21)(16,33,34,40,35,22,18,25)(28,42,43,36,44,37,31,41);(5,7)(6,9,10,11)(8,12)(13,15,18,31,20,29,35,44,19,32,34,43,17,30,16,28)(14,21,25,37,23,39,22,36,26,38,40,42,24,27,33,41);(1,3,4,2)(13,16,17,34,19,35,20,18)(14,22,24,25,26,33,23,40)(15,28,30,43,32,44,29,31)(21,36,27,37,38,41,39,42)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=i,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ci,d^a=di,f^a=fgi,d^b=dg128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35499);