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 | |
4072 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,9,10,7)(4,12,13,8)(11,14,16,15)(17,18,22,26)(19,27,30,23)(20,31,29,24)(21,25,28,32)(33,34,39,37)(35,41,45,40)(36,46,42,38)(43,44,47,48);(2,6)(7,9)(8,12)(14,15)(17,19,22,30)(18,23,26,27)(20,21,29,28)(24,25,31,32)(33,35)(34,40)(36,43)(37,41)(38,44)(39,45)(42,47)(46,48);(2,6)(3,10)(8,12)(11,16)(17,20,22,29)(18,24,26,31)(19,28,30,21)(23,32,27,25)(33,35)(34,41)(36,47)(37,40)(38,44)(39,45)(42,43)(46,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,32)(25,31)(26,27)(33,36)(34,38)(35,43)(37,46)(39,42)(40,44)(41,48)(45,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,32)(29,31)(33,37,39,34)(35,41,45,40)(36,46,42,38)(43,48,47,44);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,21,22,28)(18,25,26,32)(19,29,30,20)(23,31,27,24)(33,38,39,46)(34,42,37,36)(35,44,45,48)(40,47,41,43)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8034587);