Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574070 |
| 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 | |
1968 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,6)(3,7,4,8)(9,10)(11,15)(12,24)(13,18)(14,19)(16,23)(17,20)(21,22);(2,6)(7,8)(9,11,14,12)(10,15,19,24)(13,22,20,23)(16,18,21,17);(1,3,5,4)(2,7,6,8)(9,12,13,23,14,11,20,22)(10,16,17,24,19,21,18,15);(1,3,5,4)(2,7,6,8)(9,11,14,12)(10,15,19,24)(13,22,20,23)(16,18,21,17);(1,4,5,3)(2,8,6,7)(9,10)(11,21)(12,16)(13,17)(14,19)(15,22)(18,20)(23,24)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fg,d^2=fi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6493023);