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 | |
4307 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)(3,10,4,7)(5,8,13,15)(11,16,12,14)(17,18,22,25)(19,23,26,30)(20,21,24,29)(27,28,31,32)(33,34,40,44)(35,45,48,41)(36,39,38,42)(37,47,46,43);(1,3,6,4)(2,7,9,10)(5,11,13,12)(8,14,15,16)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32)(33,35,40,48)(34,41,44,45)(36,46,38,37)(39,47,42,43);(5,13)(8,15)(11,12)(14,16)(19,26)(23,30)(27,31)(28,32)(33,36,40,38)(34,42,44,39)(35,46,48,37)(41,43,45,47);(1,2)(3,7)(4,10)(5,8)(6,9)(11,14)(12,16)(13,15)(17,20)(18,21)(19,27)(22,24)(23,28)(25,29)(26,31)(30,32)(33,37)(34,43)(35,36)(38,48)(39,41)(40,46)(42,45)(44,47);(1,4,6,3)(2,7,9,10)(5,12,13,11)(8,14,15,16)(17,21)(18,24)(19,28)(20,25)(22,29)(23,31)(26,32)(27,30)(33,38,40,36)(34,42,44,39)(35,46,48,37)(41,47,45,43);(1,5,6,13)(2,8,9,15)(3,11,4,12)(7,14,10,16)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32)(33,39,40,42)(34,36,44,38)(35,47,48,43)(37,41,46,45)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8800644);