Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519083 |
| 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 | |
2107 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,17)(12,13)(14,19)(15,24)(16,20)(18,22)(21,23);(2,6)(7,8)(9,11,16,18)(10,17,20,22)(12,14,21,15)(13,19,23,24);(1,3,5,4)(2,7,6,8)(9,12,15,11,16,21,14,18)(10,17,19,13,20,22,24,23);(1,4,5,3)(2,8,6,7)(9,13,14,17,16,23,15,22)(10,18,24,21,20,11,19,12);(2,6)(7,8)(9,14,16,15)(10,19,20,24)(11,21,18,12)(13,17,23,22)Polycyclic presentation:
2^9|b^2=i,c^2=fg,d^2=fgi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,d^c=di,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7469022);