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 | |
4245 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)(3,8)(4,10)(5,15)(6,16)(7,9)(11,14)(12,13)(17,18,23,26)(19,25,22,24)(20,29,30,21)(27,28,31,32)(33,34,38,39)(35,36,43,40)(37,45,46,41)(42,47,48,44);(1,3)(2,8)(4,7)(5,11)(6,12)(9,10)(13,16)(14,15)(17,19,23,22)(18,24,26,25)(20,27,30,31)(21,28,29,32)(34,39)(36,40)(41,45)(44,47);(2,9)(3,4)(5,16)(12,14)(20,30)(21,29)(27,31)(28,32)(33,35,38,43)(34,40,39,36)(37,42,46,48)(41,47,45,44);(1,4,7,3)(2,10,9,8)(5,13,16,11)(6,14,15,12)(17,20)(18,21)(19,27)(22,31)(23,30)(24,28)(25,32)(26,29)(33,36,38,40)(34,35,39,43)(37,44,46,47)(41,42,45,48);(1,5)(2,6)(3,11)(4,13)(7,16)(8,12)(9,15)(10,14)(17,21)(18,20)(19,28)(22,32)(23,29)(24,27)(25,31)(26,30)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,45)(46,48);(1,6)(2,5)(3,12)(4,14)(7,15)(8,11)(9,16)(10,13)(17,22,23,19)(18,25,26,24)(20,31,30,27)(21,32,29,28)(33,37,38,46)(34,41,39,45)(35,42,43,48)(36,44,40,47)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=hi,d^2=h,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, 8034555);