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 | |
4362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(11,17)(14,18)(19,23)(22,24)(25,26,31,28)(27,30,35,32)(29,33,37,36)(34,38,40,39);(2,5)(6,7)(9,10)(11,18)(12,20)(13,16)(14,17)(15,21)(19,22)(23,24)(29,37)(33,36)(34,40)(38,39);(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,17,18)(12,15,21,20)(19,22,23,24)(25,27,31,35)(26,32,28,30)(29,34,37,40)(33,39,36,38);(1,3,4,8)(2,6,5,7)(9,10)(11,14)(12,15)(13,16)(17,18)(19,22)(20,21)(23,24)(25,28,31,26)(27,30,35,32)(29,36,37,33)(34,38,40,39);(9,11,13,17)(10,14,16,18)(12,19,21,23)(15,22,20,24)(25,29)(26,33)(27,34)(28,36)(30,38)(31,37)(32,39)(35,40);(9,12)(10,15)(11,19)(13,21)(14,22)(16,20)(17,23)(18,24)(25,30)(26,27)(28,35)(29,38)(31,32)(33,34)(36,40)(37,39)Polycyclic presentation:
2^9|a^2=h,c^2=ghi,d^2=gh,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8664218);