Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514172 |
| 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 | |
1894 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)(3,6)(4,5)(7,8)(9,10,15,19)(11,16,22,34)(12,24,17,31)(13,27,28,26)(14,29,32,18)(20,38,39,21)(23,37,30,35)(25,40,36,33);(2,5)(6,8)(9,11)(10,16)(12,25)(13,21)(14,30)(15,22)(17,36)(18,37)(19,34)(20,27)(23,32)(24,40)(26,39)(28,38)(29,35)(31,33);(1,3,4,7)(2,6,5,8)(9,12)(10,13)(11,20)(14,27)(15,17)(16,33)(18,31)(19,28)(21,35)(22,39)(23,25)(24,29)(26,32)(30,36)(34,40)(37,38);(9,13,15,28)(10,17,19,12)(11,21,22,38)(14,31,32,24)(16,25,34,36)(18,26,29,27)(20,35,39,37)(23,40,30,33);(2,5)(6,8)(11,22)(16,34)(20,39)(21,38)(23,30)(25,36)(33,40)(35,37)Polycyclic presentation:
2^9|a^2=i,c^2=f,d^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=cgi,d^b=dh,e^b=ei,g^b=gi,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, 7422112);