Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32888 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4406 conjugacy classes of subgroups.
| 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)(14,17)(15,19)(18,20);(6,9)(10,12);(5,7,8,11)(6,10,9,12)(13,14)(15,20)(16,18)(17,19);(15,19)(17,20)Polycyclic presentation:
2^9|a^2=h,c^2=e,f^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,f^c=fi,g^c=gi,f^d=fi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72317710, #72317716, #72317736, … |
| maximal normal subgroup of | #72317710, #72317716, #72317736, … |
| maximal quotient of | #72317877 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299421);