Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667836 |
| 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×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1855 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)(9,10)(11,15)(12,16)(13,14)(17,18,20,21)(19,22,24,23);(1,3)(2,5)(4,8)(6,7)(17,19)(18,22)(20,24)(21,23);(2,6)(5,7)(10,14)(15,16)(18,21)(19,23)(22,24);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,18,20,21)(19,22,24,23);(3,8)(5,7)(9,12,13,11)(10,16,14,15)(18,21)(19,23)(22,24)Polycyclic presentation:
2^9|a^2=i,d^2=fg,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72305431, #72305467, #72305476 |
| maximal normal subgroup of | #72305431, #72305467, #72305476 |
| maximal quotient of | #72305641, #72305723 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6648849);