Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535823 |
| 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 | C2×#127 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1297 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)(3,7,4,6)(9,10,13,15)(11,17,18,14)(12,16,20,22)(19,23,24,21)(25,26)(27,31)(28,37)(29,32)(30,33)(34,35)(36,39)(38,40);(1,3,5,4)(2,6,8,7)(9,11,13,18)(10,14,15,17)(12,19,20,24)(16,21,22,23)(25,27)(26,31)(28,38)(29,35)(30,36)(32,34)(33,39)(37,40);(1,3,5,4)(2,7,8,6)(10,15)(11,18)(16,22)(19,24)(25,28,37,26,30,32,29,33)(27,34,40,39,36,38,35,31);(1,4,5,3)(2,6,8,7)(12,20)(16,22)(19,24)(21,23)(25,29,30,37)(26,32,33,28)(27,35,36,40)(31,34,39,38);(9,12,13,20)(10,16,15,22)(11,19,18,24)(14,21,17,23)(27,36)(31,39)(34,38)(35,40)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=fgi,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ei,d^c=di,e^d=ehi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6468096);