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 | |
5162 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)(10,12)(14,16)(17,18,22,25)(19,20,27,23)(21,24,30,32)(26,28,29,31);(2,6)(3,8)(11,15)(14,16)(17,19)(18,23)(20,25)(21,29)(22,27)(24,28)(26,30)(31,32);(1,3)(2,4)(5,8)(6,7)(9,10,13,12)(11,14,15,16)(21,30)(24,32)(26,29)(28,31);(1,3)(2,4)(5,8)(6,7)(9,11,13,15)(10,14,12,16)(17,20,22,23)(18,19,25,27)(21,28,30,31)(24,26,32,29);(1,4)(2,3)(5,7)(6,8)(9,12,13,10)(11,16,15,14)(17,21,22,30)(18,24,25,32)(19,26,27,29)(20,28,23,31);(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=gi,c^2=hi,d^2=gh,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8510771);