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 | #44881 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1702 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,15)(12,13)(14,16)(19,28)(23,29)(24,32)(30,31);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,29)(20,27)(21,26)(22,25)(23,28)(24,30)(31,32);(1,3,5,4)(2,7,8,6)(10,13)(14,15)(17,19)(18,23)(20,31)(21,28)(22,30)(24,27)(25,32)(26,29);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,14,13,15)(17,19)(18,24)(20,30)(21,28)(22,31)(23,27)(25,29)(26,32);(1,4,5,3)(2,7,8,6)(17,20,21,22)(18,25,26,27)(19,30,28,31)(23,24,29,32)Polycyclic presentation:
2^9|b^2=f,c^2=f,d^2=fgi,e^2=fi,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, 7460914);