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 | |
4027 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)(4,14)(5,9)(6,10)(7,11)(12,16)(13,15)(17,18,23,27)(19,24,29,22)(20,26,21,25)(28,32,30,31)(33,34)(35,42)(36,39)(37,44)(38,40)(41,46)(43,47)(45,48);(1,3,7,12)(2,8,11,16)(4,15,9,6)(5,10,14,13)(17,19,23,29)(18,24,27,22)(20,30,21,28)(25,32,26,31)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(3,12)(6,15)(8,16)(10,13)(17,20,23,21)(18,25,27,26)(19,28,29,30)(22,32,24,31)(33,34,39,36)(35,42,38,40)(37,41,46,44)(43,48,45,47);(1,4)(2,5)(3,6)(7,9)(8,13)(10,16)(11,14)(12,15)(17,21,23,20)(18,26,27,25)(19,28,29,30)(22,32,24,31)(33,36,39,34)(35,40,38,42)(37,44,46,41)(43,47,45,48);(1,5)(2,9)(3,13)(4,11)(6,16)(7,14)(8,15)(10,12)(17,19)(18,22)(20,28)(21,30)(23,29)(24,27)(25,32)(26,31)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,6)(2,10)(3,4)(5,16)(7,15)(8,14)(9,12)(11,13)(17,22)(18,19)(20,31)(21,32)(23,24)(25,30)(26,28)(27,29)(33,38,39,35)(34,40,36,42)(37,45,46,43)(41,47,44,48)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=h,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8314166);