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 | |
4206 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,10,4,7)(5,8,14,13)(11,16,12,15)(17,18)(19,21)(20,25)(22,32)(23,26)(24,30)(27,29)(28,31)(33,34,38,42)(35,40,45,39)(36,46,41,37)(43,44,47,48);(1,3,6,4)(2,7,9,10)(5,11,14,12)(8,15,13,16)(17,19,23,22)(18,21,26,32)(20,28,30,27)(24,29,25,31)(33,35,38,45)(34,39,42,40)(36,43,41,47)(37,44,46,48);(1,2,6,9)(3,7,4,10)(5,13,14,8)(11,16,12,15)(17,20,23,30)(18,24,26,25)(19,27,22,28)(21,31,32,29)(33,35,38,45)(34,39,42,40)(36,47,41,43)(37,48,46,44);(1,2)(3,7)(4,10)(5,8)(6,9)(11,15)(12,16)(13,14)(17,20)(18,25)(19,28)(21,31)(22,27)(23,30)(24,26)(29,32)(33,36,38,41)(34,37,42,46)(35,43,45,47)(39,44,40,48);(1,4,6,3)(2,7,9,10)(5,12,14,11)(8,15,13,16)(17,21)(18,19)(20,29)(22,26)(23,32)(24,28)(25,27)(30,31)(33,35)(34,40)(36,47)(37,44)(38,45)(39,42)(41,43)(46,48);(1,5,6,14)(2,8,9,13)(3,11,4,12)(7,15,10,16)(17,22,23,19)(18,21,26,32)(20,27,30,28)(24,29,25,31)(33,37)(34,41)(35,44)(36,42)(38,46)(39,47)(40,43)(45,48)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=gi,d^2=hi,e^2=ghi,f^2=gh,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, 8924515);