Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7654 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6,8,11,9,12,7,10)(13,14,17,18,16,19,15,20);(6,10,12,11)(14,18)(15,17)(19,20);(1,2,3,4)(14,19)(18,20);(5,7,9,8)(6,10,12,11)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=fi,d^2=h,e^2=hi,g^2=hi,b^a=be,c^a=cf,e^a=eh,e^b=ei,g^b=gi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 112161);