Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514148 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2600 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)(3,7)(4,6)(5,8)(9,10)(11,22)(12,26)(13,21)(14,20)(15,19)(16,25)(17,28)(18,24)(23,34)(27,38)(29,37)(30,36)(31,35)(32,39)(33,40);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,27)(13,24)(14,25)(15,30)(17,33)(18,19)(20,22)(21,36)(23,28)(26,32)(29,39)(31,40)(34,35)(37,38);(9,12)(10,17)(11,23)(13,28)(14,29)(15,31)(16,32)(18,34)(19,26)(20,35)(21,37)(22,38)(24,39)(25,40)(27,30)(33,36);(9,11)(10,18)(12,23)(13,24)(14,25)(15,30)(16,21)(17,34)(19,22)(20,36)(26,38)(27,31)(28,39)(29,40)(32,37)(33,35);(2,6)(5,7)(9,13,14,15)(10,19,20,21)(11,24,25,30)(12,28,29,31)(16,18,22,36)(17,26,35,37)(23,39,40,27)(32,34,38,33)Polycyclic presentation:
2^9|b^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,g^c=gi,h^c=hi,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421527);