Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #184773 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
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 | C27 | C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
11470 conjugacy classes of subgroups; only the 450 normal subgroups are shown.
This group has 11470 conjugacy classes of subgroups — too many to draw. Only its 450 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) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,11,12,9)(8,10)(13,14,15,18,16,17,19,20);(5,7)(6,9)(8,12)(10,11);(2,4);(14,17)(18,20)Polycyclic presentation:
2^9|a^2=g,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,g^b=gh104 conjugacy classes — view the full character table.
| maximal subgroup of | #72218291, #72219244, #72234219, … |
| maximal normal subgroup of | #72218291, #72219244, #72234219, … |
| maximal quotient of | C2×#184773, #72235793, #72237555, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 91969);