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 | |
4255 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,4,10,8)(5,13,14,9)(11,15,16,12)(17,18)(19,24)(20,26)(21,30)(22,29)(23,25)(27,31)(28,32)(33,34,39,40)(35,41,36,38)(37,45,47,42)(43,46,44,48);(2,7)(3,10)(9,13)(11,16)(17,19)(18,24)(20,23)(21,29)(22,30)(25,26)(27,32)(28,31)(34,40)(35,36)(42,45)(43,44);(2,7)(3,10)(9,13)(11,16)(18,25)(19,20)(27,30)(28,29)(33,35,39,36)(34,38,40,41)(37,43,47,44)(42,46,45,48);(1,3,6,10)(2,8,7,4)(5,11,14,16)(9,15,13,12)(17,20,23,19)(18,26,25,24)(21,29,31,28)(22,27,32,30)(33,36,39,35)(34,41,40,38)(37,44,47,43)(42,48,45,46);(1,4,6,8)(2,3,7,10)(5,12,14,15)(9,11,13,16)(17,21)(18,27)(19,29)(20,28)(22,26)(23,31)(24,32)(25,30)(33,37,39,47)(34,42,40,45)(35,43,36,44)(38,46,41,48);(1,5,6,14)(2,9,7,13)(3,11,10,16)(4,12,8,15)(17,22)(18,28)(19,30)(20,27)(21,26)(23,32)(24,31)(25,29)(33,38,39,41)(34,35,40,36)(37,46,47,48)(42,43,45,44)Polycyclic presentation:
2^9|a^2=g,c^2=ghi,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8064214);