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 | |
4261 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)(3,7)(4,12)(5,6)(8,13)(9,10)(11,15)(14,16)(17,18,22,26)(19,20,24,23)(21,25,29,32)(27,28,31,30)(33,34,38,42)(35,40,45,39)(36,46,41,37)(43,44,47,48);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,35,38,45)(34,39,42,40)(36,43,41,47)(37,44,46,48);(2,6)(3,10)(4,13)(14,15)(19,24)(20,23)(21,29)(25,32)(33,35,38,45)(34,39,42,40)(36,47,41,43)(37,48,46,44);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,19)(18,23)(20,26)(21,27)(22,24)(25,30)(28,32)(29,31)(33,36,38,41)(34,37,42,46)(35,43,45,47)(39,44,40,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,20,22,23)(18,24,26,19)(21,28,29,30)(25,31,32,27)(33,35)(34,40)(36,47)(37,44)(38,45)(39,42)(41,43)(46,48);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,21,22,29)(18,25,26,32)(19,27,24,31)(20,28,23,30)(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=hi,c^2=hi,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, 8924893);