Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| 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 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1856 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)(3,7)(4,8)(5,6)(9,10,13,15)(11,14,12,16);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,18)(19,21)(20,22)(23,24);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,14,15,16)(18,21)(19,24)(22,23);(1,4,5,3)(2,8,6,7)(18,21)(19,24)(22,23);(18,22)(21,23)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=fg,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6355167);