Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4924 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,12)(13,14)(15,16)(17,18,21,24)(19,22,27,25)(20,28,29,23)(26,31,32,30);(2,3)(6,7)(10,14)(11,15)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,29)(31,32);(1,2,5,3)(4,6,8,7)(9,11,13,15)(10,12,14,16)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31);(1,3,5,2)(4,7,8,6)(9,12)(10,15)(11,14)(13,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,29)(30,32);(1,4,5,8)(2,6,3,7);(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)Polycyclic presentation:
2^9|a^2=gi,c^2=g,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8015224);