Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994043 |
| 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×D8 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(6,7)(9,10,12,14)(11,13,16,15)(17,18,21,24)(19,22,25,29)(20,27,28,23)(26,32,31,30);(1,2,4,5)(3,7,8,6)(19,25)(20,28)(22,29)(23,27);(10,13)(11,16)(14,15)(17,19)(18,22)(20,26)(21,25)(23,30)(24,29)(27,32)(28,31);(1,3,4,8)(2,6,5,7)(10,14)(13,15)(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=hi,b^2=f,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7075532);