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 | |
4205 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,8)(3,6,10,9)(4,7,13,12)(11,14,16,15)(17,18)(19,20)(21,23)(22,24)(25,28)(26,27)(29,31)(30,32)(33,34)(35,42)(36,44)(37,39)(38,40)(41,46)(43,47)(45,48);(1,2)(3,9)(4,7)(5,8)(6,10)(11,15)(12,13)(14,16)(17,19,23,25)(18,20,21,28)(22,26,31,32)(24,27,29,30)(33,35,39,38)(34,40,37,42)(36,45,46,43)(41,48,44,47);(3,10)(6,9)(11,16)(14,15)(18,21)(19,25)(24,29)(26,32)(34,37)(40,42)(41,44)(47,48);(1,2)(3,6)(4,12)(5,8)(7,13)(9,10)(11,15)(14,16)(17,20)(18,19)(21,25)(22,30)(23,28)(24,32)(26,29)(27,31)(33,36)(34,41)(35,43)(37,44)(38,45)(39,46)(40,47)(42,48);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,21,23,18)(19,20,25,28)(22,29,31,24)(26,27,32,30)(33,37,39,34)(35,40,38,42)(36,44,46,41)(43,47,45,48);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(17,22)(18,24)(19,26)(20,27)(21,29)(23,31)(25,32)(28,30)(33,38,39,35)(34,40,37,42)(36,43,46,45)(41,48,44,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8495564);