Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574075 |
| 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 | |
1992 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)(9,10,13,17)(11,20,12,15)(14,19,21,18)(16,23,24,22);(2,6)(7,8)(9,11)(10,15)(12,13)(14,22)(16,18)(17,20)(19,24)(21,23);(1,3,5,4)(2,7,6,8)(9,11,21,23,13,12,14,22)(10,16,19,15,17,24,18,20);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,15,17,20)(14,23,21,22)(16,19,24,18);(1,4,5,3)(2,8,6,7)(9,10,13,17)(11,16,12,24)(14,18,21,19)(15,22,20,23)Polycyclic presentation:
2^9|a^2=f,c^2=fg,d^2=f,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6493028);