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 |
| prime radical | #42434 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4257 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,18)(12,13)(14,15)(16,21)(17,22)(19,20)(23,24)(26,32)(27,28)(34,37)(35,36);(2,5)(6,7)(10,15)(12,22)(16,18)(19,24)(25,26)(27,33)(28,30)(29,37)(31,32)(34,39)(35,38)(36,40);(1,2)(3,6)(4,5)(7,8)(25,27,31,28)(26,33,32,30)(29,35,39,36)(34,40,37,38);(1,2)(3,6)(4,5)(7,8)(9,11,14,21)(10,16,15,18)(12,19,22,24)(13,23,17,20)(25,28,31,27)(26,33,32,30)(29,36,39,35)(34,40,37,38);(1,3,4,8)(2,6,5,7)(9,12)(10,17)(11,19)(13,15)(14,22)(16,20)(18,23)(21,24)(25,29)(26,34)(27,35)(28,36)(30,38)(31,39)(32,37)(33,40);(9,13,14,17)(10,12,15,22)(11,20,21,23)(16,24,18,19)(25,30,31,33)(26,28,32,27)(29,38,39,40)(34,36,37,35)Polycyclic presentation:
2^9|a^2=i,c^2=hi,d^2=gi,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, 8416066);