Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514196 |
| 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 | C2×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1661 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,13,15,16)(18,23)(24,29)(25,31)(27,32);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,18,21,23)(19,24,22,29)(20,27,28,32)(25,30,31,26);(1,3,5,4)(2,7,6,8)(10,14)(11,15)(17,19,21,22)(18,24,23,29)(20,26,28,30)(25,32,31,27);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(17,20)(18,25)(19,26)(21,28)(22,30)(23,31)(24,32)(27,29);(2,6)(7,8)(20,28)(25,31)(26,30)(27,32)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=f,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7433021);