Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129405 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C2×#2408 | C2 |
| LCS | G ▷ C25 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
2550 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 2550 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) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,4)(9,10)(11,15,16,13)(12,14);(2,4)(5,6)(7,8)(10,13)(14,15);(6,8)(10,13)(14,15)Polycyclic presentation:
2^9|a^2=g,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,c^b=cf,g^b=gh,g^c=gh,g^d=gi,g^e=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72232755, #72232941, #72233099, … |
| maximal normal subgroup of | #72232755, #72232941, #72233099, … |
| maximal quotient of | #72232754, #72233901, #72234179, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54906);