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 | |
1872 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,28)(14,18)(15,20)(16,26)(17,31)(19,33)(21,37)(22,25)(23,27)(24,34)(29,40)(30,35)(32,36)(38,39);(2,5)(6,8)(9,11,15,23)(10,12,20,27)(13,29,32,21)(14,16,33,22)(17,24,35,39)(18,26,19,25)(28,40,36,37)(30,38,31,34);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,18)(13,24)(14,25)(15,27)(17,34)(19,23)(20,22)(21,28)(26,33)(29,36)(30,37)(31,40)(32,39)(35,38);(9,13)(10,17)(11,21)(12,24)(14,30)(15,32)(16,34)(18,28)(19,36)(20,35)(22,38)(23,29)(25,37)(26,40)(27,39)(31,33);(2,5)(6,8)(9,14,15,33)(10,18,20,19)(11,22,23,16)(12,25,27,26)(13,30,32,31)(17,28,35,36)(21,38,29,34)(24,37,39,40)Polycyclic presentation:
2^9|b^2=i,c^2=f,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=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, 7422116);