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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2067 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)(11,17)(12,13)(14,16)(15,21)(18,24)(19,23)(20,22);(1,3,5,4)(2,6,8,7)(9,11)(10,17)(12,20)(13,22)(14,24)(15,23)(16,18)(19,21);(9,12)(10,18)(11,14)(13,21)(15,17)(16,19)(20,23)(22,24);(1,3,5,4)(2,7,8,6)(9,13,16,23)(10,19,14,12)(11,22,18,15)(17,21,24,20);(1,4,5,3)(2,6,8,7)(9,14)(10,16)(11,12)(13,17)(15,21)(18,19)(20,22)(23,24)Polycyclic presentation:
2^9|b^2=f,d^2=fi,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, 6485856);