Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6369150 |
| 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 | |
1561 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)(4,5)(8,15)(9,16)(10,14)(11,13)(17,18,21,26)(19,28,29,23)(20,25,31,32)(22,24,30,27);(1,2,6,10)(3,8,12,16)(4,7,5,14)(9,13,15,11)(17,19)(18,23)(20,22)(21,29)(24,32)(25,27)(26,28)(30,31);(1,3)(2,8)(4,13)(5,11)(6,12)(7,15)(9,14)(10,16)(18,24)(19,29)(22,30)(23,25)(26,27)(28,32);(1,3)(2,9)(4,11)(5,13)(6,12)(7,16)(8,14)(10,15)(17,20)(18,25)(19,30)(21,31)(22,29)(23,24)(26,32)(27,28);(1,4,6,5)(2,7,10,14)(3,11,12,13)(8,9,16,15)(18,26)(19,29)(20,31)(24,27)Polycyclic presentation:
2^9|a^2=i,b^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,c^b=ci,d^b=df,e^b=ei,f^b=fh,e^c=eh,g^c=gi,e^d=ei,f^d=fh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6276363);