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 | |
4119 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)(3,10)(4,7)(5,8)(6,9)(11,16)(12,14)(13,15)(17,18)(19,26)(20,23)(21,28)(22,24)(25,30)(27,31)(29,32)(33,34,39,38)(35,42,37,40)(36,41,44,46)(43,48,45,47);(1,3,7,5)(2,8,4,10)(6,11,14,13)(9,15,12,16)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(36,44)(41,46)(43,45)(47,48);(6,14)(9,12)(11,13)(15,16)(17,18,23,20)(19,26,22,24)(21,25,30,28)(27,32,29,31)(33,35,39,37)(34,40,38,42)(36,43,44,45)(41,47,46,48);(1,4,7,2)(3,8,5,10)(6,12,14,9)(11,15,13,16)(17,20,23,18)(19,24,22,26)(21,28,30,25)(27,31,29,32)(33,36,39,44)(34,41,38,46)(35,43,37,45)(40,47,42,48);(1,5,7,3)(2,8,4,10)(6,13,14,11)(9,15,12,16)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,37,39,35)(34,42,38,40)(36,45,44,43)(41,48,46,47);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,22,23,19)(18,24,20,26)(21,29,30,27)(25,31,28,32)(33,38,39,34)(35,42,37,40)(36,46,44,41)(43,48,45,47)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=h,d^2=ghi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8095208);