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 | |
4225 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)(3,7)(4,12)(5,6)(8,13)(9,10)(11,15)(14,16)(17,18,22,25)(19,20,27,23)(21,24,29,31)(26,28,32,30)(33,34)(35,41)(36,44)(37,40)(38,43)(39,46)(42,47)(45,48);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,34,37,40)(35,38,43,41)(36,39,46,44)(42,47,45,48);(2,6)(3,10)(8,12)(11,16)(21,29)(24,31)(26,32)(28,30)(33,35,37,43)(34,38,40,41)(36,42,46,45)(39,47,44,48);(1,2)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(17,20,22,23)(18,19,25,27)(21,30,29,28)(24,32,31,26)(33,35)(34,38)(36,45)(37,43)(39,48)(40,41)(42,46)(44,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,21,22,29)(18,24,25,31)(19,26,27,32)(20,28,23,30)(33,34)(35,41)(36,39)(37,40)(38,43)(42,48)(44,46)(45,47);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,32)(30,31)(33,36,37,46)(34,39,40,44)(35,42,43,45)(38,47,41,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=hi,d^2=ghi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,e^d=egi,f^d=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9575558);