Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573643 |
| 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 |
| prime radical | #23954 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ 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)(3,7)(4,6)(5,8)(9,10)(11,16)(12,13)(14,19)(15,20)(17,22)(18,21)(23,24);(1,3,5,4)(2,6,8,7)(9,11,15,23)(10,16,20,24)(12,18,17,19)(13,21,22,14);(9,12,15,17)(10,11,20,23)(13,18,22,19)(14,24,21,16);(1,3,5,4)(2,7,8,6)(9,13)(10,17)(11,21)(12,20)(14,23)(15,22)(16,19)(18,24);(1,4,5,3)(2,6,8,7)(9,10)(11,12)(13,16)(14,18)(15,20)(17,23)(19,21)(22,24)Polycyclic presentation:
2^9|b^2=fi,c^2=i,d^2=f,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,g^c=gi,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6485855);