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 | |
4020 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,11,12,8)(4,5,10,9)(13,14,16,15)(17,18)(19,24)(20,21)(22,23)(25,30)(26,29)(27,31)(28,32)(33,34,40,39)(35,44,38,41)(36,43,37,42)(45,47,46,48);(2,7)(3,12)(4,10)(14,15)(17,18)(19,26)(20,21)(22,23)(24,29)(25,30)(27,28)(31,32)(33,35,40,38)(34,41,39,44)(36,46,37,45)(42,48,43,47);(2,7)(4,10)(8,11)(13,16)(18,23)(20,30)(24,26)(27,32)(33,36,40,37)(34,42,39,43)(35,45,38,46)(41,47,44,48);(1,3,6,12)(2,8,7,11)(4,13,10,16)(5,15,9,14)(17,19)(18,24)(20,27)(21,31)(22,29)(23,26)(25,28)(30,32)(33,37,40,36)(34,43,39,42)(35,46,38,45)(41,48,44,47);(1,4,6,10)(2,9,7,5)(3,13,12,16)(8,14,11,15)(17,20)(18,25)(19,27)(21,23)(22,30)(24,28)(26,31)(29,32)(33,38,40,35)(34,44,39,41)(36,46,37,45)(42,48,43,47);(1,5,6,9)(2,10,7,4)(3,14,12,15)(8,13,11,16)(17,21,22,25)(18,20,23,30)(19,28,29,31)(24,32,26,27)(33,39,40,34)(35,44,38,41)(36,43,37,42)(45,48,46,47)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=ghi,d^2=g,e^2=g,f^2=h,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, 8514646);