Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | #15315 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2376 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,7,8,6)(9,10,12,14)(11,13,15,16)(17,18,22,23)(19,21,20,24)(25,26)(27,31)(28,30)(29,32);(3,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,19,22,20)(18,21,23,24)(27,32)(29,31);(2,5)(6,7)(11,15)(13,16)(18,23)(19,20)(25,26,28,30)(27,29,32,31);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,14)(13,15)(17,20,22,19)(18,21,23,24)(25,26)(27,29)(28,30)(31,32);(1,3)(2,6)(4,8)(5,7)(9,11)(10,13)(12,15)(14,16)(17,21)(18,19)(20,23)(22,24)(25,27)(26,29)(28,32)(30,31)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=fgi,d^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4126205);