Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6366542 |
| 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 | |
1580 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)(3,9)(4,10)(5,11)(6,8)(7,12)(13,16)(14,15)(17,18)(19,30)(20,31)(21,28)(22,27)(23,29)(24,26)(25,32);(2,8)(4,9)(5,6)(10,15)(11,12)(14,16)(17,19)(18,24)(20,29)(21,31)(22,26)(23,25)(27,32)(28,30);(1,3,7,13)(2,9,12,16)(4,11,14,8)(5,15,6,10)(17,20)(18,25)(19,29)(21,32)(22,30)(23,24)(26,28)(27,31);(1,4)(2,10)(3,11)(5,9)(6,16)(7,14)(8,13)(12,15)(17,20)(18,26)(19,27)(21,30)(22,32)(23,24)(25,28)(29,31);(1,5,7,6)(2,8,12,11)(3,10,13,15)(4,9,14,16)(17,21,23,22)(18,27,29,28)(19,31,25,26)(20,30,24,32)Polycyclic presentation:
2^9|c^2=hi,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=ehi,f^b=fh,e^c=eh,g^c=gi,f^d=fh65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6273798);