Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6367198 |
| 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 | |
1597 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,11)(3,12,14,7)(4,9,6,10)(8,15,16,13)(17,18)(19,30)(20,26)(21,28)(22,27)(23,29)(24,31)(25,32);(1,3)(2,7)(4,15)(5,14)(6,13)(8,9)(10,16)(11,12)(17,19,23,25)(18,24,29,20)(21,26,22,31)(27,32,28,30);(1,3)(2,8)(4,13)(5,14)(6,15)(7,9)(10,12)(11,16)(17,20)(18,25)(19,29)(21,32)(22,30)(23,24)(26,27)(28,31);(1,4,5,6)(2,9,11,10)(3,13,14,15)(7,16,12,8)(17,20)(18,26)(19,28)(21,30)(22,32)(23,24)(25,27)(29,31);(1,4,5,6)(2,10,11,9)(3,13,14,15)(7,8,12,16)Polycyclic presentation:
2^9|a^2=hi,b^2=h,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,c^b=ch,d^b=df,e^b=ehi,f^b=fh,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, 6274708);