Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6358586 |
| 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 | |
1554 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,7,12)(3,13,15,16)(4,9,6,11)(5,10,8,14)(17,18)(19,23)(20,27)(21,29)(22,25)(24,26)(28,31)(30,32);(1,3)(2,8)(4,10)(5,12)(6,14)(7,15)(9,16)(11,13)(17,19,22,21)(18,23,25,29)(20,26,28,30)(24,31,32,27);(1,4,7,6)(2,9,12,11)(3,14,15,10)(5,16,8,13)(17,20)(18,24)(19,26)(21,30)(22,28)(23,31)(25,32)(27,29);(1,5)(2,10)(3,9)(4,16)(6,13)(7,8)(11,15)(12,14);(1,4,7,6)(2,11,12,9)(3,10,15,14)(5,16,8,13)(17,19,22,21)(18,23,25,29)(20,26,28,30)(24,31,32,27)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,c^b=ch,d^b=df,f^b=fh,g^c=gi,f^d=fh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6265915);