Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519063 |
| 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 | |
2090 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)(4,8)(5,6)(9,10)(11,21)(12,27)(13,29)(14,30)(15,20)(16,25)(17,18)(19,31)(22,39)(23,28)(24,34)(26,35)(32,40)(33,36)(37,38);(2,6)(7,8)(9,11)(10,16)(12,28)(13,23)(14,24)(15,25)(17,36)(18,22)(19,34)(20,21)(26,31)(27,32)(29,40)(30,35)(33,38)(37,39);(1,3,5,4)(2,7,6,8)(9,12,14,18,15,29,31,38)(10,17,19,27,20,37,30,13)(11,22,26,28,25,33,24,40)(16,32,35,36,21,23,34,39);(1,4,5,3)(2,8,6,7)(9,13,14,17,15,27,31,37)(10,18,19,29,20,38,30,12)(11,23,26,39,25,32,24,36)(16,33,35,40,21,22,34,28);(2,6)(7,8)(9,14,15,31)(10,19,20,30)(11,24,25,26)(12,18,29,38)(13,17,27,37)(16,34,21,35)(22,40,33,28)(23,36,32,39)Polycyclic presentation:
2^9|c^2=fgi,d^2=fgi,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,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7468970);