Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996671 |
| 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 | #59815 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1740 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,4)(3,7,8,6)(9,10,14,11)(12,15,13,16)(17,18,21,23)(19,22,24,20);(1,3,5,8)(2,6,4,7)(10,11)(12,13)(17,19,21,24)(18,22,23,20);(1,2,5,4)(3,6,8,7)(9,11,14,10)(12,15,13,16)(17,20,19,23,21,22,24,18);(1,4,5,2)(3,6,8,7)(9,12,14,13)(10,15,11,16);(9,13,14,12)(10,15,11,16)(17,19,21,24)(18,20,23,22)Polycyclic presentation:
2^9|a^2=fi,b^2=fh,c^2=fg,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7094608);