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 | |
4332 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,7,4)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26,31,37)(27,36,30,32)(28,34,39,33)(29,35,40,38);(1,3)(2,6)(4,8)(5,7)(11,17)(15,21)(18,23)(22,24)(25,27,31,30)(26,32,37,36)(28,35,39,38)(29,33,40,34);(1,2)(3,4)(5,8)(6,7)(10,14)(15,21)(16,19)(22,24)(25,28,31,39)(26,33,37,34)(27,35,30,38)(29,32,40,36);(1,4)(2,7)(3,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,28)(26,34)(27,35)(29,32)(30,38)(31,39)(33,37)(36,40);(1,4)(2,3)(5,6)(7,8)(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24)(25,29)(26,35)(27,34)(28,32)(30,33)(31,40)(36,39)(37,38);(1,2)(3,6)(4,7)(5,8)(9,12,13,20)(10,16,14,19)(11,18,17,23)(15,22,21,24)(25,30,31,27)(26,36,37,32)(28,38,39,35)(29,34,40,33)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=g,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9625654);