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 | |
4319 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)(9,10,14,13)(11,15,17,19)(12,16,20,22)(18,23,21,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,39)(36,40)(37,38);(1,3,5,8)(2,6,4,7)(10,13)(11,17)(12,20)(23,24)(27,33)(28,36)(29,31)(32,40);(3,8)(6,7)(9,11)(10,15)(12,21)(13,19)(14,17)(16,24)(18,20)(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)(9,12)(10,16)(11,18)(13,22)(14,20)(15,23)(17,21)(19,24)(25,28,30,36)(26,32,34,40)(27,35,33,37)(29,38,31,39);(1,4,5,2)(3,6,8,7)(9,13,14,10)(11,19,17,15)(12,22,20,16)(18,24,21,23)(25,29,30,31)(26,33,34,27)(28,38,36,39)(32,37,40,35)Polycyclic presentation:
2^9|a^2=g,b^2=hi,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, 8929125);