Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1702 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,4,8,6)(10,14)(12,15)(17,18)(19,20)(21,24)(22,23)(27,31)(30,32);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,24)(21,23)(25,26,29,28)(27,30,31,32);(9,11)(10,12)(13,16)(14,15)(17,20)(18,23)(19,24)(21,22)(25,26,29,28)(27,30,31,32);(1,4,5,6)(2,3,7,8)(9,11)(10,15)(12,14)(13,16)(17,20)(18,19)(21,22)(23,24)(25,27)(26,30)(28,32)(29,31);(1,3)(2,4)(5,8)(6,7)(9,12,13,15)(10,16,14,11)(17,18)(19,22)(20,23)(21,24)(25,28,29,26)(27,32,31,30)Polycyclic presentation:
2^9|a^2=fgh,b^2=fghi,c^2=fghi,d^2=fgh,e^2=fgi,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, 4835947);