Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6353301 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2723 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)(3,11)(4,9)(5,8)(6,10)(7,12)(13,15)(14,16)(17,18,23,19)(20,28,30,32)(21,25,22,27)(24,31,26,29);(1,3,6,13)(2,7,10,16)(4,12,5,14)(8,11,9,15)(17,19,23,18)(20,26,30,24)(21,27,22,25)(28,31,32,29);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,11,16,15)(17,20)(18,24)(19,26)(21,29)(22,31)(23,30)(25,32)(27,28);(1,2)(3,7)(4,8)(5,9)(6,10)(11,12)(13,16)(14,15)(17,21,23,22)(18,25,19,27)(20,29,30,31)(24,32,26,28);(1,4,6,5)(2,9,10,8)(3,12,13,14)(7,15,16,11)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=h,d^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,f^b=fh,g^b=gi,f^c=fi,g^c=gi,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6260657);