Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667652 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #36096 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2199 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(17,18)(19,24)(20,23)(21,22);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,20,21)(18,22,23,24);(3,8)(5,7)(9,11,12,16)(10,13,14,15)(18,22)(19,21)(23,24);(2,6)(5,7)(18,23)(22,24);(10,14)(13,15)Polycyclic presentation:
2^9|b^2=i,c^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^c=gi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72239614, #72305466, #72305477, … |
| maximal normal subgroup of | #72239614, #72305466, #72305477, … |
| maximal quotient of | #72239773, #72306161, #72306183 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6660230);