Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 2097152 |
| 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 | C23×#41 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2592 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,4,7,6)(9,10)(11,15)(12,13)(14,16)(18,22)(19,24)(26,30)(28,31);(1,3)(2,6)(4,8)(5,7)(9,11)(10,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,22)(26,30)(27,32);(17,19)(18,23)(20,22)(21,24)(25,26)(27,31)(28,32)(29,30);(1,3)(2,4)(5,7)(6,8)(9,12,13,10)(11,15,16,14)(17,19)(18,20)(21,24)(22,23)(25,27,29,32)(26,31,30,28);(1,4)(2,7)(3,8)(5,6)(9,11)(10,15)(12,14)(13,16)(17,20)(18,24)(19,22)(21,23)(25,28,29,31)(26,32,30,27)Polycyclic presentation:
2^9|a^2=fi,d^2=gi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4802837);