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 | |
4871 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,8)(5,6)(11,17)(15,18)(19,23)(22,24)(25,26,30,33)(27,29,35,31)(28,36,37,32)(34,38,40,39);(1,2,5,6)(3,7,8,4)(9,10)(11,15)(12,20)(13,14)(16,21)(17,18)(19,24)(22,23)(25,27,30,35)(26,31,33,29)(28,34,37,40)(32,38,36,39);(1,3)(2,4)(5,8)(6,7)(10,14)(15,18)(16,20)(22,24)(28,37)(32,36)(34,40)(38,39);(1,4)(2,3)(5,7)(6,8)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,28,30,37)(26,32,33,36)(27,34,35,40)(29,39,31,38);(1,2)(3,4)(5,6)(7,8)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,24)(21,23)(25,29)(26,27)(28,38)(30,31)(32,40)(33,35)(34,36)(37,39);(1,4)(2,3)(5,7)(6,8)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,27)(26,29)(28,34)(30,35)(31,33)(32,39)(36,38)(37,40)Polycyclic presentation:
2^9|a^2=gh,b^2=g,d^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8731936);