Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500908 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2064 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8)(3,4,7,6)(9,10,14,35,16,21,15,20)(11,17,25,37,26,39,24,38)(12,27,36,18,30,23,29,40)(13,31,33,19,34,22,32,28);(1,3,5,7)(2,6,8,4)(9,11,16,26)(10,17,21,39)(12,28,30,19)(13,23,34,27)(14,25,15,24)(18,33,40,32)(20,38,35,37)(22,36,31,29);(1,4,5,6)(2,7,8,3)(9,12)(10,18)(11,22)(13,17)(14,29)(15,36)(16,30)(19,25)(20,23)(21,40)(24,28)(26,31)(27,35)(32,37)(33,38)(34,39);(2,8)(3,7)(9,13)(10,19)(11,23)(12,17)(14,32)(15,33)(16,34)(18,25)(20,22)(21,28)(24,40)(26,27)(29,37)(30,39)(31,35)(36,38);(9,14,16,15)(10,20,21,35)(11,24,26,25)(12,29,30,36)(13,32,34,33)(17,37,39,38)(18,23,40,27)(19,22,28,31)Polycyclic presentation:
2^9|a^2=fgi,b^2=fi,c^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,h^b=hi,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7411039);