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 | |
4277 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,18)(12,15)(13,21)(14,17)(16,22)(19,20)(23,24)(25,26,32,36)(27,37,39,30)(28,38,40,33)(29,35,31,34);(3,7)(6,8)(9,11,14,15)(10,12,17,18)(13,19,22,23)(16,20,21,24)(25,27,32,39)(26,30,36,37)(28,35,40,34)(29,33,31,38);(1,2)(3,6)(4,5)(7,8)(9,10,14,17)(11,12,15,18)(13,21,22,16)(19,24,23,20)(25,28,32,40)(26,33,36,38)(27,34,39,35)(29,37,31,30);(9,10)(11,12)(13,16)(14,17)(15,18)(19,20)(21,22)(23,24)(25,29,32,31)(26,34,36,35)(27,33,39,38)(28,37,40,30);(1,3,4,7)(2,6,5,8)(9,12)(10,15)(11,17)(13,20)(14,18)(16,23)(19,21)(22,24)(25,30)(26,27)(28,29)(31,40)(32,37)(33,34)(35,38)(36,39);(1,2)(3,6)(4,5)(7,8)(9,13,14,22)(10,16,17,21)(11,19,15,23)(12,20,18,24)(25,31,32,29)(26,35,36,34)(27,38,39,33)(28,30,40,37)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=g,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8837723);