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 | |
4066 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,4)(3,8,5,9)(6,12,14,10)(11,16,13,15)(17,18,23,27)(19,26,21,24)(20,22,31,25)(28,30,29,32)(34,37)(39,42)(40,45)(46,48);(1,3)(2,8)(4,9)(5,7)(6,13)(10,16)(11,14)(12,15)(17,19,23,21)(18,24,27,26)(20,28,31,29)(22,32,25,30)(36,44)(40,45)(41,47)(46,48);(2,4)(3,5)(10,12)(11,13)(17,20,23,31)(18,25,27,22)(19,28,21,29)(24,30,26,32)(33,34,38,37)(35,39,43,42)(36,40,44,45)(41,46,47,48);(1,4,7,2)(3,9,5,8)(6,12,14,10)(11,16,13,15)(17,21,23,19)(18,26,27,24)(20,28,31,29)(22,32,25,30)(33,35,38,43)(34,39,37,42)(36,41,44,47)(40,46,45,48);(1,5,7,3)(2,9,4,8)(6,13,14,11)(10,16,12,15)(17,20)(18,22)(19,29)(21,28)(23,31)(24,30)(25,27)(26,32)(33,36,38,44)(34,40,37,45)(35,41,43,47)(39,46,42,48);(1,6,7,14)(2,10,4,12)(3,11,5,13)(8,15,9,16)(17,22)(18,20)(19,30)(21,32)(23,25)(24,29)(26,28)(27,31)(33,37,38,34)(35,42,43,39)(36,45,44,40)(41,48,47,46)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=hi,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,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, 8680064);