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 | |
4051 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)(4,10)(5,11)(6,16)(7,9)(12,15)(13,14)(17,18,22,25)(19,23,27,21)(20,24,30,32)(26,31,28,29)(33,34)(35,43)(36,41)(37,39)(38,42)(40,44)(45,46)(47,48);(1,3,7,6)(2,8,9,16)(4,12,14,11)(5,10,15,13)(17,19)(18,23)(20,26)(21,25)(22,27)(24,31)(28,30)(29,32)(33,35)(34,40)(36,38)(37,43)(39,44)(41,47)(42,46)(45,48);(2,9)(3,6)(4,14)(5,15)(17,19,22,27)(18,23,25,21)(20,28,30,26)(24,29,32,31)(33,36,39,45)(34,41,37,46)(35,38,44,48)(40,47,43,42);(1,4)(2,10)(3,12)(5,16)(6,11)(7,14)(8,15)(9,13)(17,19)(18,23)(20,28)(21,25)(22,27)(24,29)(26,30)(31,32)(33,37,39,34)(35,43,44,40)(36,41,45,46)(38,47,48,42);(1,5,7,15)(2,11,9,12)(3,13,6,10)(4,8,14,16)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,32)(27,28)(33,36)(34,41)(35,38)(37,46)(39,45)(40,47)(42,43)(44,48);(1,6,7,3)(2,8,9,16)(4,11,14,12)(5,10,15,13)(17,21,22,23)(18,19,25,27)(20,29,30,31)(24,26,32,28)(33,38)(34,42)(35,36)(37,47)(39,48)(40,46)(41,43)(44,45)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=ghi,d^2=g,e^2=h,f^2=i,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, 8934492);