Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500740 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1720 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)(11,16)(12,25)(13,28)(14,18)(15,19)(17,32)(20,21)(22,35)(23,34)(24,36)(26,39)(27,31)(29,30)(33,37)(38,40);(1,3,5,7)(2,6,8,4)(9,11,15,24)(10,16,19,36)(12,26,29,37)(13,21,31,40)(14,23,32,22)(17,35,18,34)(20,27,38,28)(25,39,30,33);(1,4,5,6)(2,7,8,3)(9,12)(10,13)(11,20)(14,27)(15,29)(16,33)(17,30)(18,25)(19,31)(21,34)(22,37)(23,26)(24,38)(28,32)(35,40)(36,39);(2,8)(3,7)(9,13)(10,12)(11,21)(14,30)(15,31)(16,26)(17,27)(18,28)(19,29)(20,34)(22,39)(23,33)(24,40)(25,32)(35,38)(36,37);(9,14,15,32)(10,17,19,18)(11,22,24,23)(12,27,29,28)(13,30,31,25)(16,34,36,35)(20,37,38,26)(21,39,40,33)Polycyclic presentation:
2^9|a^2=f,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,g^a=gi,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, 7409258);