Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500908 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 | |
2715 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)(9,10,12,18,14,19,13,17)(11,15,21,23,22,24,20,16);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,21)(13,20)(14,22)(16,17)(18,23)(19,24);(9,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,24)(22,23);(2,6)(3,8)(9,11)(10,16)(12,20)(13,21)(14,22)(15,17)(18,24)(19,23);(9,12,14,13)(10,17,19,18)(11,20,22,21)(15,23,24,16)Polycyclic presentation:
2^9|a^2=g,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,h^b=hi,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410459);