Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393940 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1758 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(6,7)(9,10)(11,12)(13,14,16,17)(15,19,20,18);(5,6,8,7)(14,17)(15,18)(19,20);(2,4)(5,7,8,6)(13,15,16,20)(14,18,17,19);(10,12)Polycyclic presentation:
2^9|a^2=efi,b^2=ei,c^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72311810, #72311814, #72311845, … |
| maximal normal subgroup of | #72311810, #72311814, #72311845, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303111);