Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2846 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C24 |
| commutator | C23 | C22×C42 |
| socle | C24 | C23×C4 |
| prime radical | C26 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4882 conjugacy classes of subgroups; only the 770 normal subgroups are shown.
This group has 4882 conjugacy classes of subgroups — too many to draw. Only its 770 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,18,16,17,19,20,15);(2,4)(13,15,18,14,17,16,20,19);(6,8);(13,16,18,19,17,15,20,14)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di176 conjugacy classes — view the full character table.
| maximal subgroup of | #72247723, #72253046, #97×#120, … |
| maximal normal subgroup of | #97×#120, #72279725, #72279750, … |
| maximal quotient of | #72286354, #72313241, #72320097, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261977);