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 | |
4151 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,6)(3,10,5,8)(4,12,13,9)(11,15,14,16)(17,18,23,27)(19,24,28,30)(20,25,29,32)(21,22,26,31)(34,37)(35,36)(42,45)(43,44);(1,3,7,5)(2,8,6,10)(4,11,13,14)(9,15,12,16)(17,19,23,28)(18,24,27,30)(20,22,29,31)(21,25,26,32)(33,34,39,37)(35,41,36,40)(38,42,46,45)(43,48,44,47);(4,13)(9,12)(11,14)(15,16)(19,28)(20,29)(24,30)(25,32)(33,35,39,36)(34,40,37,41)(38,43,46,44)(42,47,45,48);(1,4,7,13)(2,9,6,12)(3,11,5,14)(8,15,10,16)(17,20)(18,25)(19,22)(21,30)(23,29)(24,26)(27,32)(28,31)(33,36,39,35)(34,41,37,40)(38,44,46,43)(42,48,45,47);(1,5,7,3)(2,8,6,10)(4,14,13,11)(9,15,12,16)(17,21)(18,22)(19,25)(20,30)(23,26)(24,29)(27,31)(28,32)(33,37,39,34)(35,41,36,40)(38,45,46,42)(43,48,44,47);(1,6,7,2)(3,10,5,8)(4,9,13,12)(11,15,14,16)(17,22)(18,26)(19,29)(20,28)(21,27)(23,31)(24,32)(25,30)(33,38,39,46)(34,42,37,45)(35,43,36,44)(40,47,41,48)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=h,d^2=g,e^2=g,f^2=g,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, 8619332);