Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993875 |
| 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 | |
1730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,6)(4,7)(9,10,15,18,14,11,13,19)(12,17,22,16,24,21,23,20);(1,2,5,6)(3,4,8,7)(9,11,14,10)(12,16,24,20)(13,18,15,19)(17,22,21,23);(1,3)(2,7)(4,6)(5,8)(9,12,14,24)(10,16,11,20)(13,22,15,23)(17,19,21,18);(2,6)(3,8)(9,12)(10,17)(11,21)(13,23)(14,24)(15,22)(16,18)(19,20);(1,4,5,7)(2,3,6,8)(10,11)(16,20)(17,21)(18,19)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7089194);