Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #195954 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C43 |
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 | C2×#2867 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2982 conjugacy classes of subgroups; only the 490 normal subgroups are shown.
This group has 2982 conjugacy classes of subgroups — too many to draw. Only its 490 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 — 4 generators on 16 points:
(2,4)(6,8)(9,10,12,15,13,14,16,11);(9,11,12,14,13,15,16,10);(1,2,3,4);(5,6,7,8)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=f,d^2=g,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,e^b=ei,h^b=hi104 conjugacy classes — view the full character table.
| Fitting subgroup of | #72228324, #72258697 |
| maximal subgroup of | #72225944, #72225962, #72226231, … |
| maximal normal subgroup of | #72225944, #72225962, #72226231, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300976);