Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500740 |
| 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 | |
1784 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)(3,4)(5,7)(6,8)(9,10,14,20)(11,22,17,16)(12,24,18,23)(13,19,15,21);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,21)(13,23)(14,17)(15,24)(18,19)(20,22);(9,12,14,18)(10,17,20,11)(13,16,15,22)(19,24,21,23);(2,7)(3,8)(9,11,14,17)(10,18,20,12)(13,24,15,23)(16,21,22,19);(1,4,5,6)(2,3,7,8)(9,13,14,15)(10,19,20,21)(11,23,17,24)(12,16,18,22)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=i,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=df,h^b=hi,g^c=gi,h^c=hi,e^d=ei,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409745);