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 | |
4152 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,11)(3,8,13,6)(4,9,14,15)(5,10,12,16)(17,18)(19,20)(21,29)(22,26)(23,24)(25,32)(27,30)(28,31)(33,34,39,42)(35,40,45,37)(36,46,38,41)(43,47,44,48);(1,3)(2,8)(4,5)(6,11)(7,13)(9,10)(12,14)(15,16)(17,19)(18,23)(20,26)(21,27)(22,24)(25,30)(28,29)(31,32)(33,35,39,45)(34,40,42,37)(36,44,38,43)(41,47,46,48);(1,4)(2,9)(3,12)(5,13)(6,10)(7,14)(8,16)(11,15)(17,20)(18,24)(19,26)(21,28)(22,23)(25,31)(27,29)(30,32)(33,36,39,38)(34,41,42,46)(35,43,45,44)(37,47,40,48);(1,5,7,12)(2,10,11,16)(3,4,13,14)(6,15,8,9)(17,21,22,32)(18,25,26,29)(19,27,24,31)(20,28,23,30)(33,37)(34,35)(36,47)(38,48)(39,40)(41,43)(42,45)(44,46);(1,4)(2,9)(3,5)(6,16)(7,14)(8,10)(11,15)(12,13)(17,20)(18,19)(21,30)(22,23)(24,26)(25,31)(27,29)(28,32)(33,38,39,36)(34,41,42,46)(35,44,45,43)(37,47,40,48);(1,6,7,8)(2,3,11,13)(4,10,14,16)(5,9,12,15)(17,19)(18,23)(20,26)(21,31)(22,24)(25,28)(27,32)(29,30)(33,34)(35,40)(36,41)(37,45)(38,46)(39,42)(43,48)(44,47)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=gh,d^2=hi,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9140042);