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 | |
5015 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)(4,6)(5,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,26)(19,23,27,32)(20,24,30,29)(21,25,31,28);(1,3,4,8)(2,5,6,7)(9,10,12,14)(11,13,16,15)(19,27)(20,30)(23,32)(24,29);(1,2,4,6)(3,5,8,7)(9,10)(11,13)(12,14)(15,16)(17,19)(18,23)(20,28)(21,24)(22,27)(25,30)(26,32)(29,31);(1,3)(2,5)(4,8)(6,7)(9,11,12,16)(10,13,14,15);(1,3)(2,5)(4,8)(6,7)(17,20)(18,24)(19,28)(21,23)(22,30)(25,27)(26,29)(31,32);(1,2)(3,5)(4,6)(7,8)(17,21)(18,25)(19,29)(20,23)(22,31)(24,27)(26,28)(30,32)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=h,d^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8109564);