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 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1947 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)(11,23)(12,26)(13,17)(14,22)(15,21)(16,20)(18,31)(19,32)(24,39)(25,34)(27,38)(28,35)(29,37)(30,36)(33,40);(2,5)(6,8)(9,11,15,13)(10,17,21,23)(12,27,30,40)(14,25,16,32)(18,35,37,39)(19,20,34,22)(24,29,28,31)(26,33,36,38);(9,12)(10,18)(11,24)(13,28)(14,31)(15,30)(16,29)(17,33)(19,35)(20,26)(21,37)(22,36)(23,38)(25,40)(27,32)(34,39);(1,3,4,7)(2,6,5,8)(9,13,15,11)(10,19,21,34)(12,28,30,24)(14,32,16,25)(17,22,23,20)(18,35,37,39)(26,33,36,38)(27,29,40,31);(2,5)(6,8)(9,14,15,16)(10,20,21,22)(11,25,13,32)(12,29,30,31)(17,34,23,19)(18,36,37,26)(24,27,28,40)(33,35,38,39)Polycyclic presentation:
2^9|b^2=i,d^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=cg,d^b=dh,g^b=gi,e^c=ei,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, 7423737);