Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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×#553 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1376 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)(3,8)(4,6)(5,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(27,34)(29,32)(36,39)(37,38);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,26)(27,36)(28,33)(29,37)(30,35)(31,40)(32,38)(34,39);(1,3,5,4)(2,6,7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,27)(26,32)(28,38)(29,35)(30,34)(31,37)(33,39)(36,40);(10,16)(14,17)(15,22)(21,23)(25,28,30,31)(26,33,35,40)(27,37,34,38)(29,39,32,36);(1,4,5,3)(2,6,7,8)(11,19)(14,17)(18,24)(21,23)(25,29,30,32)(26,34,35,27)(28,39,31,36)(33,38,40,37)Polycyclic presentation:
2^9|b^2=f,c^2=fgi,d^2=i,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6629414);