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 | |
4018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,8,11,5)(4,6,9,15)(12,13,16,14)(17,18,22,25)(19,23,27,21)(20,28,31,24)(26,30,29,32)(33,34)(35,42)(36,46)(37,45)(38,40)(39,41)(43,47)(44,48);(1,3)(2,8)(4,12)(5,10)(6,13)(7,11)(9,16)(14,15)(17,19)(18,23)(20,29)(21,25)(22,27)(24,30)(26,31)(28,32)(33,35,40,45)(34,37,38,42)(36,43,39,44)(41,47,46,48);(3,11)(5,8)(12,16)(13,14)(17,20,22,31)(18,24,25,28)(19,26,27,29)(21,30,23,32)(33,36,40,39)(34,41,38,46)(35,43,45,44)(37,47,42,48);(1,4)(2,6)(3,12)(5,14)(7,9)(8,13)(10,15)(11,16)(17,21)(18,19)(20,30)(22,23)(24,26)(25,27)(28,29)(31,32)(33,37)(34,35)(36,47)(38,45)(39,48)(40,42)(41,43)(44,46);(1,5)(2,3)(4,14)(6,12)(7,8)(9,13)(10,11)(15,16)(17,20)(18,24)(19,26)(21,30)(22,31)(23,32)(25,28)(27,29)(33,38,40,34)(35,37,45,42)(36,41,39,46)(43,48,44,47);(1,6)(2,9)(3,13)(4,10)(5,12)(7,15)(8,16)(11,14)(17,19)(18,23)(20,26)(21,25)(22,27)(24,32)(28,30)(29,31)(33,39,40,36)(34,41,38,46)(35,44,45,43)(37,47,42,48)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=h,e^2=g,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, 8494665);