Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6366870 |
| 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 | |
1574 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,7)(3,5)(8,15)(9,10)(11,14)(12,16)(17,18,23,27)(19,28,24,31)(20,30,21,25)(22,29,32,26);(1,2)(3,7)(4,12)(5,9)(6,10)(8,11)(13,15)(14,16)(17,19)(18,21)(20,27)(22,25)(23,24)(26,31)(28,29)(30,32);(1,3,6,5)(2,7,10,9)(4,11,13,14)(8,15,16,12)(17,20)(18,24)(19,27)(21,23)(22,31)(25,26)(28,32)(29,30);(1,4,6,13)(2,8,10,16)(3,11,5,14)(7,15,9,12)(17,21)(18,25)(19,29)(20,23)(22,31)(24,26)(27,30)(28,32);(1,3,6,5)(2,9,10,7)(4,11,13,14)(8,12,16,15)Polycyclic presentation:
2^9|a^2=i,c^2=hi,d^2=hi,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,d^b=df,e^b=ehi,f^b=fh,g^c=gi,f^d=fh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6274388);