Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 | #2537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1459 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)(11,15)(12,14)(13,16)(17,18,21,22)(19,23,27,24)(20,28,30,25)(26,29,32,31);(2,6)(7,8)(11,16)(13,15)(17,19,21,27)(18,23,22,24)(20,29,30,31)(25,26,28,32);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,26)(21,27)(22,23)(25,29)(28,31)(30,32);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,20,21,30)(18,25,22,28)(19,26,27,32)(23,31,24,29);(1,4,5,3)(2,7,6,8)(19,27)(23,24)(26,32)(29,31)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgi,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7442695);