Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884873 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1859 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,6)(3,7,8,5)(18,21)(19,22)(23,24);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,15)(13,16)(17,18,20,21)(19,23,24,22);(3,8)(5,7)(9,11,14,10)(12,15,13,16)(17,19)(18,22)(20,24)(21,23);(2,6)(5,7)(9,12,14,13)(10,15,11,16)(17,18,20,21)(19,22,24,23);(9,13,14,12)(10,15,11,16)(17,18,20,21)(19,22,24,23)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=fh,d^2=fhi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6899998);