Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129405 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C25 | C22×C4 |
| Frattini, prime radical, cosocle | C23×D4 | C23 |
| LCS | G ▷ C25 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
1698 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 1698 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) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(13,14)(15,19,20,17)(16,18);(5,6,8,10)(7,11,12,9)(14,17)(18,19);(2,4)(5,7,8,12)(6,9,10,11)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,g^b=gh,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72268841, #72281618, #72281641, … |
| maximal normal subgroup of | #72268841, #72281618, #72281641, … |
| maximal quotient of | #72268852, #72319570 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56621);