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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1626 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,3)(6,7)(10,14)(11,16)(17,18,22,24)(19,20,23,21)(25,26,28,30)(27,31,32,29);(1,2,5,3)(4,6,8,7)(9,10,13,14)(11,12,16,15)(17,19)(18,21)(20,24)(22,23)(27,32)(29,31);(9,11)(10,12)(13,16)(14,15)(17,20,22,21)(18,23,24,19)(25,26)(27,29)(28,30)(31,32);(1,3,5,2)(4,6,8,7)(9,11)(10,15)(12,14)(13,16)(17,19)(18,20)(21,24)(22,23)(25,27)(26,29)(28,32)(30,31);(1,4,5,8)(2,6,3,7)(9,12,13,15)(10,16,14,11)(17,21,22,20)(18,23,24,19)Polycyclic presentation:
2^9|a^2=fi,b^2=fh,c^2=h,d^2=gi,e^2=f,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, 4802695);