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 | |
4090 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,7,11,5)(4,8,14,16)(10,15,12,13)(17,18)(19,25)(20,24)(21,22)(23,26)(27,32)(28,31)(29,30)(33,34)(35,42)(36,41)(37,40)(38,43)(39,44)(45,48)(46,47);(1,3)(2,7)(4,10)(5,9)(6,11)(8,15)(12,14)(13,16)(19,26)(23,25)(27,28)(31,32)(33,35,39,37)(34,40,44,42)(36,45,47,38)(41,43,46,48);(1,3,6,11)(2,7,9,5)(4,12,14,10)(8,13,16,15)(17,19,22,26)(18,23,21,25)(20,28,30,27)(24,32,29,31)(33,36,39,47)(34,41,44,46)(35,45,37,38)(40,43,42,48);(1,3)(2,7)(4,12)(5,9)(6,11)(8,13)(10,14)(15,16)(17,20,22,30)(18,24,21,29)(19,27,26,28)(23,31,25,32)(33,37,39,35)(34,42,44,40)(36,45,47,38)(41,43,46,48);(1,4)(2,8)(3,10)(5,13)(6,14)(7,15)(9,16)(11,12)(17,19)(18,25)(20,28)(21,23)(22,26)(24,31)(27,30)(29,32)(33,38,39,45)(34,43,44,48)(35,36,37,47)(40,46,42,41);(1,5,6,7)(2,3,9,11)(4,13,14,15)(8,10,16,12)(17,21,22,18)(19,23,26,25)(20,29,30,24)(27,31,28,32)(33,34)(35,40)(36,46)(37,42)(38,43)(39,44)(41,47)(45,48)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=gi,d^2=gh,e^2=g,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, 8933934);