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 | |
4081 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)(3,8)(4,9)(5,10)(6,7)(11,15)(12,13)(14,16)(17,18,23,26)(19,24,21,22)(20,25,28,32)(27,31,30,29)(33,34)(35,38)(36,41)(37,42)(39,43)(40,46)(44,45)(47,48);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,14)(13,16)(17,19,23,21)(18,24,26,22)(20,27,28,30)(25,31,32,29)(33,35,39,46)(34,40,43,38)(36,44,42,48)(37,45,41,47);(2,6)(3,5)(4,12)(15,16)(19,21)(20,28)(22,24)(25,32)(33,36,39,42)(34,41,43,37)(35,44,46,48)(38,45,40,47);(1,4,7,12)(2,9,6,13)(3,11,5,14)(8,15,10,16)(17,20)(18,25)(19,27)(21,30)(22,29)(23,28)(24,31)(26,32)(33,36)(34,37)(35,44)(38,47)(39,42)(40,45)(41,43)(46,48);(1,5,7,3)(2,10,6,8)(4,11,12,14)(9,15,13,16)(17,21,23,19)(18,22,26,24)(20,27,28,30)(25,31,32,29)(33,37)(34,42)(35,45)(36,43)(38,44)(39,41)(40,48)(46,47);(1,6,7,2)(3,10,5,8)(4,13,12,9)(11,16,14,15)(17,22)(18,19)(20,29)(21,26)(23,24)(25,27)(28,31)(30,32)(33,38)(34,35)(36,47)(37,44)(39,40)(41,48)(42,45)(43,46)Polycyclic presentation:
2^9|a^2=hi,b^2=ghi,c^2=g,d^2=h,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8914338);