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 | |
4243 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)(19,23)(20,28)(21,29)(22,24)(25,30)(26,32)(27,31)(33,34,38,40)(35,41,43,36)(37,45,46,39)(42,44,48,47);(2,6)(7,9)(8,12)(14,15)(17,19,22,25)(18,23,24,30)(20,26,29,31)(21,27,28,32)(33,35)(34,36)(37,42)(38,43)(39,44)(40,41)(45,47)(46,48);(2,6)(3,10)(8,12)(11,16)(18,24)(19,25)(20,29)(27,32)(33,34,38,40)(35,36,43,41)(37,39,46,45)(42,44,48,47);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,34)(35,36)(37,39)(38,40)(41,43)(42,44)(45,46)(47,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,20)(18,21)(19,26)(22,29)(23,27)(24,28)(25,31)(30,32)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,48)(46,47);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,21)(18,20)(19,27)(22,28)(23,26)(24,29)(25,32)(30,31)(33,37,38,46)(34,39,40,45)(35,42,43,48)(36,44,41,47)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=ghi,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, 8034435);