Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10046 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C23 | #379 |
| prime radical | #34976 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6,8,11,9,7,12,10)(14,18)(15,16)(19,20);(1,2)(3,4)(5,7)(6,9)(8,10)(11,12)(13,14)(15,20)(16,18)(17,19);(2,4)(13,14)(15,18)(16,20)(17,19)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=dh,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239478, #72239481, #72239491 |
| maximal normal subgroup of | #72239478, #72239481, #72239491 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18721);