Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628842 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4314 conjugacy classes of subgroups; only the 826 normal subgroups are shown.
This group has 4314 conjugacy classes of subgroups — too many to draw. Only its 826 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 16 points:
(2,4)(6,8)(10,14)(11,12)(15,16);(1,2,3,4);(5,6,7,8);(9,10)(11,14)(12,16)(13,15);(9,11,13,12)(10,14,15,16)Polycyclic presentation:
2^9|b^2=f,c^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,h^a=hi,h^d=hi104 conjugacy classes — view the full character table.
| Fitting subgroup of | #72228321, #72258670 |
| maximal subgroup of | #72225944, #72225960, #72226222, … |
| maximal normal subgroup of | #72225944, #72225960, #72226222, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6537191);