Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993887 |
| 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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2191 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(3,7)(6,8)(9,10,15,18)(11,13,19,14)(12,17,22,20)(16,23,24,21);(2,5)(3,7)(9,11,13,18,15,19,14,10)(12,20,21,16,22,17,23,24);(1,2)(3,6)(4,5)(7,8)(9,12,15,22)(10,16,18,24)(11,17,19,20)(13,21,14,23);(2,5)(6,8)(9,12)(10,17)(11,16)(13,23)(14,21)(15,22)(18,20)(19,24);(1,3,4,7)(2,6,5,8)(10,18)(11,19)(16,24)(17,20)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,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, 7050826);