Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518944 |
| 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×#550 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1535 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)(11,13)(12,14)(15,16)(17,18,23,20)(19,28,29,25)(21,30,32,26)(22,31,24,27);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,25)(20,28)(21,22)(23,29)(24,32)(26,31)(27,30);(1,3,5,4)(2,7,6,8)(10,14)(11,15)(17,20,23,18)(19,28,29,25)(21,30,32,26)(22,27,24,31);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(17,21,22,19,23,32,24,29)(18,26,31,25,20,30,27,28);(2,6)(7,8)(17,22,23,24)(18,27,20,31)(19,21,29,32)(25,30,28,26)Polycyclic presentation:
2^9|a^2=i,c^2=f,d^2=fh,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7517386);