Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993887 |
| 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 | |
1889 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)(3,5)(4,6)(7,8)(9,10,14,17,15,18,13,20)(11,19,21,23,22,24,16,12);(1,2)(3,7)(4,6)(5,8)(9,11,14,21,15,22,13,16)(10,12,20,24,18,23,17,19);(1,3)(2,5)(4,8)(6,7)(9,12)(10,16)(11,20)(13,19)(14,24)(15,23)(17,22)(18,21);(3,8)(5,7)(9,12)(10,11)(13,24)(14,19)(15,23)(16,20)(17,21)(18,22);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=g,b^2=fi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6910265);