Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667676 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26,30,33)(27,34,36,39)(28,32,29,37)(31,38,40,35);(3,4)(6,7)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22);(1,3,5,4)(2,6,8,7)(25,27)(26,31)(28,35)(29,38)(30,36)(32,39)(33,40)(34,37);(10,15)(14,17)(16,23)(22,24)(25,28,30,29)(26,32,33,37)(27,35,36,38)(31,39,40,34);(1,4,5,3)(2,7,8,6)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(26,33)(31,40)(32,37)(34,39)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=h,d^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,g^c=gi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6755449);