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 | |
4168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,5)(3,9,10,7)(4,12,14,8)(11,15,13,16)(19,25)(23,26)(27,31)(30,32)(33,34,38,40)(35,36,43,39)(37,45,47,41)(42,48,46,44);(2,5)(7,9)(8,12)(15,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,32)(30,31)(33,35,38,43)(34,39,40,36)(37,46,47,42)(41,44,45,48);(3,10)(7,9)(11,13)(15,16)(18,22)(23,26)(24,28)(30,32)(34,40)(36,39)(41,45)(44,48);(1,3)(2,7)(4,13)(5,9)(6,10)(8,16)(11,14)(12,15)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,36)(34,35)(37,44)(38,39)(40,43)(41,42)(45,46)(47,48);(1,4,6,14)(2,8,5,12)(3,11,10,13)(7,15,9,16)(17,19,21,25)(18,23,22,26)(20,27,29,31)(24,30,28,32)(33,35)(34,39)(36,40)(37,42)(38,43)(41,48)(44,45)(46,47);(1,5,6,2)(3,9,10,7)(4,12,14,8)(11,16,13,15)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,37,38,47)(34,41,40,45)(35,42,43,46)(36,44,39,48)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8762366);