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 | |
2571 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)(4,6)(5,8)(9,10)(11,12)(13,28)(14,22)(15,21)(16,20)(17,25)(18,27)(19,31)(23,37)(24,32)(26,34)(29,36)(30,35)(33,39)(38,40);(1,3,4,8)(2,5,6,7)(9,11,15,25)(10,17,21,12)(13,23,30,38)(14,24,16,18)(19,26,36,33)(20,32,22,27)(28,40,35,37)(29,34,31,39);(9,12)(10,18)(11,20)(13,26)(14,27)(15,17)(16,32)(19,34)(21,24)(22,25)(23,28)(29,37)(30,33)(31,40)(35,38)(36,39);(9,13,15,30)(10,19,21,36)(11,23,25,38)(12,26,17,33)(14,31,16,29)(18,34,24,39)(20,28,22,35)(27,40,32,37);(2,6)(5,7)(9,14,15,16)(10,20,21,22)(11,24,25,18)(12,27,17,32)(13,29,30,31)(19,35,36,28)(23,34,38,39)(26,37,33,40)Polycyclic presentation:
2^9|b^2=f,d^2=i,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,e^d=ei,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421547);