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 | |
4438 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)(25,26,30,34)(27,29,33,31)(28,32,36,40)(35,38,37,39);(3,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,39)(36,40)(37,38);(1,3,5,8)(2,6,4,7)(10,14)(11,17)(12,20)(22,24)(27,33)(28,36)(29,31)(32,40);(3,8)(6,7)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,27,30,33)(26,31,34,29)(28,37,36,35)(32,38,40,39);(1,3)(2,7)(4,6)(5,8)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,28,30,36)(26,32,34,40)(27,35,33,37)(29,38,31,39);(1,4,5,2)(3,6,8,7)(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,29,30,31)(26,33,34,27)(28,38,36,39)(32,37,40,35)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8925261);