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 | |
4241 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,9,11,7)(5,12,14,8)(10,15,13,16)(17,18,23,27)(19,24,22,32)(20,21,26,25)(28,30,29,31)(33,34)(35,43)(36,41)(37,39)(38,42)(40,44)(45,46)(47,48);(2,4)(7,9)(8,12)(15,16)(17,19,23,22)(18,24,27,32)(20,29,26,28)(21,31,25,30)(33,35)(34,40)(36,38)(37,43)(39,44)(41,47)(42,46)(45,48);(1,3,6,11)(2,7,4,9)(5,13,14,10)(8,16,12,15)(17,20,23,26)(18,25,27,21)(19,28,22,29)(24,31,32,30)(33,36,39,45)(34,41,37,46)(35,38,44,48)(40,47,43,42);(1,3)(2,7)(4,9)(5,10)(6,11)(8,15)(12,16)(13,14)(17,20)(18,21)(19,29)(22,28)(23,26)(24,31)(25,27)(30,32)(33,37,39,34)(35,43,44,40)(36,41,45,46)(38,47,48,42);(1,4,6,2)(3,7,11,9)(5,12,14,8)(10,15,13,16)(17,21)(18,26)(19,30)(20,27)(22,31)(23,25)(24,29)(28,32)(33,36)(34,41)(35,38)(37,46)(39,45)(40,47)(42,43)(44,48);(1,5,6,14)(2,8,4,12)(3,10,11,13)(7,15,9,16)(17,22,23,19)(18,24,27,32)(20,28,26,29)(21,31,25,30)(33,38)(34,42)(35,36)(37,47)(39,48)(40,46)(41,43)(44,45)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=gi,d^2=ghi,e^2=hi,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, 8924536);