Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667508 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3052 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(18,21)(22,24);(1,3)(2,5)(4,8)(6,7)(9,11,12,15)(10,13,14,16)(18,22)(20,23)(21,24);(10,14)(13,16)(17,18)(19,21)(20,24)(22,23);(2,6)(5,7);(3,8)(5,7)(17,18)(19,21)(20,24)(22,23)Polycyclic presentation:
2^9|b^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72257404, #72257516, #72264075, … |
| maximal normal subgroup of | #72257404, #72257516, #72264075, … |
| maximal quotient of | #72265349 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6585160);