Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,4,8,6)(11,15)(14,16)(17,18,21,23)(19,20,24,22)(25,26)(27,31)(28,30)(29,32);(1,3)(2,6)(4,7)(5,8)(9,10,13,12)(11,14,15,16)(17,19)(18,22)(20,23)(21,24)(25,27,30,29)(26,31,28,32);(2,7)(3,8)(19,24)(20,22)(25,26,30,28)(27,32,29,31);(1,3)(2,4)(5,8)(6,7)(9,11)(10,14)(12,16)(13,15)(17,20)(18,19)(21,22)(23,24)(25,28,30,26)(27,31,29,32);(1,4)(2,8)(3,7)(5,6)(9,12,13,10)(11,16,15,14)(17,18)(19,22)(20,24)(21,23)(25,29,30,27)(26,31,28,32)Polycyclic presentation:
2^9|a^2=f,b^2=ghi,c^2=fh,d^2=fh,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2712058);