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 | |
4061 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,9)(3,7,10,6)(4,12,13,8)(11,14,16,15)(17,18,24,23)(19,28,22,25)(20,27,21,26)(29,31,30,32)(33,34,39,43)(35,44,48,40)(36,37,47,41)(38,46,45,42);(3,10)(6,7)(11,16)(14,15)(17,19,24,22)(18,25,23,28)(20,30,21,29)(26,32,27,31)(33,34)(35,40)(36,37)(38,42)(39,43)(41,47)(44,48)(45,46);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,20,24,21)(18,26,23,27)(19,29,22,30)(25,31,28,32)(33,35)(34,40)(36,45)(37,46)(38,47)(39,48)(41,42)(43,44);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,21,24,20)(18,27,23,26)(19,30,22,29)(25,32,28,31)(33,36,39,47)(34,41,43,37)(35,45,48,38)(40,42,44,46);(1,2)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,22,24,19)(18,28,23,25)(20,30,21,29)(26,32,27,31)(33,37,39,41)(34,36,43,47)(35,46,48,42)(38,40,45,44);(1,4,5,13)(2,8,9,12)(3,11,10,16)(6,14,7,15)(17,23,24,18)(19,28,22,25)(20,27,21,26)(29,32,30,31)(33,38,39,45)(34,42,43,46)(35,47,48,36)(37,40,41,44)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=ghi,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8046671);