Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518992 |
| 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×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1445 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)(4,6)(5,8)(9,10,13,12)(11,14,16,15)(17,18)(19,27)(20,23)(21,26)(22,25)(24,30)(28,31)(29,32);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,15)(13,16)(17,19)(18,24)(20,32)(21,29)(22,30)(23,31)(25,27)(26,28);(1,3,5,4)(2,7,8,6)(9,12,13,10)(11,15,16,14)(17,18,21,20,22,25,23,26)(19,28,31,27,30,32,29,24);(1,4,5,3)(2,6,8,7)(11,16)(14,15)(17,20,21,25,22,26,23,18)(19,27,31,32,30,24,29,28);(2,8)(6,7)(17,21,22,23)(18,20,25,26)(19,29,30,31)(24,32,27,28)Polycyclic presentation:
2^9|a^2=gh,b^2=fi,c^2=fhi,d^2=fg,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7510882);