Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7654 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
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 | #36614 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2967 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)(7,9,11,12)(8,10)(14,18,19,20);(5,7)(6,9,10,12)(8,11)(13,14,16,20,17,19,15,18);(2,4);(13,15,17,16)(14,18,19,20)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=hi,e^2=hi,b^a=be,c^a=cf,e^a=eh,e^b=ei,g^b=gi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72309223, #72309232, #72309239, … |
| maximal normal subgroup of | #72309223, #72309232, #72309239, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 103559);