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 | |
4162 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,11,12,8)(4,9,15,14)(5,13,16,10)(17,18,22,24)(19,25,26,20)(21,23,29,30)(27,31,32,28)(33,34,39,42)(35,40,38,36)(37,41,47,48)(43,45,46,44);(2,7)(4,15)(8,11)(10,13)(17,18)(19,20)(21,23)(22,24)(25,26)(27,28)(29,30)(31,32)(33,35,39,38)(34,40,42,36)(37,43,47,46)(41,45,48,44);(2,7)(3,12)(4,15)(5,16)(19,26)(20,25)(21,29)(23,30)(33,36,39,40)(34,35,42,38)(37,45,47,44)(41,46,48,43);(1,3)(2,8)(4,10)(5,14)(6,12)(7,11)(9,16)(13,15)(17,19,22,26)(18,20,24,25)(21,27,29,32)(23,28,30,31)(33,36)(34,35)(37,45)(38,42)(39,40)(41,46)(43,48)(44,47);(1,4)(2,9)(3,13)(5,8)(6,15)(7,14)(10,12)(11,16)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,31)(30,32)(33,37)(34,41)(35,43)(36,44)(38,46)(39,47)(40,45)(42,48);(1,5)(2,10)(3,14)(4,8)(6,16)(7,13)(9,12)(11,15)(17,21)(18,23)(19,27)(20,28)(22,29)(24,30)(25,31)(26,32)(33,38,39,35)(34,36,42,40)(37,46,47,43)(41,44,48,45)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=i,d^2=gh,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, 8936305);