Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C25 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C26 | C23 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
3746 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)(13,14)(15,17,19,20)(16,18);(2,4)(13,15)(14,17,18,20)(16,19);(6,9)(10,12);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=f,b^a=be,c^a=cf,e^a=ei,h^a=hi,e^b=ei,g^b=gi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72231722, #72231735, #72231777, … |
| maximal normal subgroup of | #72231722, #72231735, #72231777, … |
| maximal quotient of | #72264520, #72268627, #72314235 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261020);