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 | |
4889 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,7)(4,8)(5,6)(11,12)(14,15);(2,6)(3,8)(9,10,13,16)(11,14,12,15)(17,18,22,19)(20,23,28,25)(21,24,30,26)(27,31,29,32);(1,3)(2,4)(5,8)(6,7)(9,11,13,12)(10,14,16,15)(18,19)(23,25)(24,26)(31,32);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,15)(13,16)(17,19,22,18)(20,25,28,23)(21,24,30,26)(27,31,29,32);(1,4)(2,3)(5,7)(6,8)(9,12,13,11)(10,15,16,14)(17,20,22,28)(18,23,19,25)(21,29,30,27)(24,32,26,31);(17,21,22,30)(18,24,19,26)(20,27,28,29)(23,31,25,32)Polycyclic presentation:
2^9|b^2=hi,c^2=gh,d^2=gi,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8633919);