Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667836 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1250 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26,30,34)(27,31,28,32)(29,33,37,40)(35,38,36,39);(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(27,35)(28,36)(29,37)(31,38)(32,39)(33,40);(2,5)(6,8)(25,27,30,28)(26,31,34,32)(29,36,37,35)(33,39,40,38);(1,3,4,7)(2,6,5,8)(10,15)(14,17)(16,23)(22,24)(27,28)(31,32)(35,36)(38,39);(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(25,28,30,27)(26,32,34,31)(29,35,37,36)(33,38,40,39)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=g,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6611477);