Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996405 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2164 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(3,7)(6,8)(9,10,15,23,16,12,14,19)(11,13,22,17,21,24,18,20);(2,5)(3,7)(9,11,16,21)(10,17,12,20)(13,19,24,23)(14,22,15,18);(1,2)(3,6)(4,5)(7,8)(9,12)(10,16)(11,20)(13,22)(14,23)(15,19)(17,21)(18,24);(2,5)(6,8)(9,13,14,20,16,24,15,17)(10,18,19,21,12,22,23,11);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=fi,b^2=i,d^2=f,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7053026);