Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6360896 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1671 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)(3,7)(4,11)(5,14)(6,9)(8,10)(12,16)(13,15)(17,18)(19,29)(20,25)(21,27)(22,26)(23,28)(24,31)(30,32);(1,3,6,5)(2,7,9,14)(4,12,13,10)(8,11,16,15)(17,19)(18,20)(21,25)(22,30)(23,24)(26,29)(27,31)(28,32);(1,4)(2,8)(3,10)(5,12)(6,13)(7,15)(9,16)(11,14)(17,20,23,32)(18,24,28,19)(21,31,22,29)(25,27,30,26);(1,3,6,5)(2,7,9,14)(4,10,13,12)(8,15,16,11)(17,20,23,32)(18,25,28,30)(19,26,24,27)(21,29,22,31);(17,21,23,22)(18,26,28,27)(19,25,24,30)(20,29,32,31)Polycyclic presentation:
2^9|b^2=hi,c^2=h,d^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,c^b=chi,d^b=df,f^b=fh,e^c=eh,g^c=gi,f^d=fh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6268174);