Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #184792 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C23×#121 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
5914 conjugacy classes of subgroups; only the 394 normal subgroups are shown.
This group has 5914 conjugacy classes of subgroups — too many to draw. Only its 394 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) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,18)(17,20);(6,9)(10,12)(13,14)(15,19)(16,18)(17,20);(2,4)(5,7)(6,10)(8,11)(9,12)(13,15)(14,17,18,19)(16,20);(7,11)(10,12)Polycyclic presentation:
2^9|c^2=h,b^a=be,c^a=cf,h^a=hi,c^b=cg,h^b=hi,d^c=de,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72219307, #72234616, #72234797, … |
| maximal normal subgroup of | #72219307, #72234616, #72234797, … |
| maximal quotient of | #72219305, #72234620, #72234788, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289586);