Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32929 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C2×#3050 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2676 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,22)(16,20)(17,23)(18,21)(19,24)(25,28)(26,27);(5,7,8,11)(6,9,10,12)(13,15,19,26,18,25,17,16)(14,20,23,28,21,27,24,22);(6,10)(9,12)(14,21)(20,27)(22,28)(23,24);(2,4)(13,16,17,25,18,26,19,15)(14,22,24,27,21,28,23,20);(13,17,18,19)(14,23,21,24)(15,16,25,26)(20,28,27,22)Polycyclic presentation:
2^9|b^2=fg,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,h^a=hi152 conjugacy classes — view the full character table.
| maximal subgroup of | #72317193, #72317198, #72317199, … |
| maximal normal subgroup of | #72317193, #72317198, #72317199, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536917);