Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437748 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #36624 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2894 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10,12,16,13,15,21,23)(11,17,18,22,19,24,20,14);(5,6)(7,8)(9,11,13,19)(10,14,15,22)(12,20,21,18)(16,24,23,17);(1,2,3,4)(6,8);(6,8);(10,15)(14,22)(16,23)(17,24)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72312984, #72313004, #72313093, … |
| maximal normal subgroup of | #72312984, #72313004, #72313093, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350181);