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 | |
4381 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(13,21)(17,22)(19,20)(23,24)(25,26)(27,36)(28,34)(29,33)(30,32)(31,35)(37,39)(38,40);(1,3,5,8)(2,6,4,7)(9,10,15,14)(11,18,12,16)(13,22,21,17)(19,23,20,24)(25,27,31,30)(26,32,35,36)(28,37,29,38)(33,39,34,40);(9,11,15,12)(10,16,14,18)(13,19,21,20)(17,23,22,24)(25,28,31,29)(26,33,35,34)(27,37,30,38)(32,39,36,40);(2,4)(6,7)(9,12,15,11)(10,16,14,18)(13,20,21,19)(17,23,22,24)(25,29,31,28)(26,34,35,33)(27,37,30,38)(32,39,36,40);(1,3)(2,6)(4,7)(5,8)(9,13,15,21)(10,17,14,22)(11,19,12,20)(16,23,18,24)(25,26)(27,36)(28,33)(29,34)(30,32)(31,35)(37,40)(38,39);(1,4,5,2)(3,7,8,6)(9,14,15,10)(11,18,12,16)(13,22,21,17)(19,24,20,23)(25,30,31,27)(26,32,35,36)(28,38,29,37)(33,39,34,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=hi,d^2=hi,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8416156);