Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514160 |
| 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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1891 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)(3,7)(4,5)(6,8)(9,10)(11,19)(12,24)(13,27)(14,18)(15,22)(16,25)(17,28)(20,32)(21,31)(23,30)(26,34)(29,33)(35,36)(37,39)(38,40);(2,5)(6,7)(11,20)(13,28)(15,30)(16,33)(17,27)(19,32)(21,37)(22,23)(25,29)(31,39)(35,40)(36,38);(9,11)(10,15)(12,21)(13,20)(14,23)(16,31)(17,30)(18,32)(19,27)(22,28)(24,35)(25,37)(26,38)(29,36)(33,40)(34,39);(9,12,14,26)(10,16,18,33)(11,21,23,38)(13,29,28,25)(15,31,32,40)(17,34,27,24)(19,35,30,39)(20,36,22,37);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^9|d^2=i,e^2=f,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,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, 7425472);