Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996335 |
| 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 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1689 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10,15,11,14,13,12,16)(17,18,21,24)(19,22,25,29)(20,27,28,23)(26,32,31,30);(1,3,4,8)(2,5,6,7)(9,11,14,16)(10,12,13,15)(19,25)(20,28)(22,29)(23,27);(2,6)(5,7)(9,12,14,15)(10,11,13,16)(17,19,21,25)(18,22,24,29)(20,26,28,31)(23,30,27,32);(3,8)(5,7)(9,13,12,11,14,10,15,16)(19,25)(22,29)(26,31)(30,32);(17,20,21,28)(18,23,24,27)(19,26,25,31)(22,30,29,32)Polycyclic presentation:
2^9|a^2=fhi,b^2=g,c^2=hi,d^2=fgi,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7104976);