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 | |
2069 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)(11,17)(12,25)(13,23)(14,19)(15,22)(16,20)(18,29)(21,35)(24,31)(26,34)(27,38)(28,36)(30,37)(32,33)(39,40);(2,6)(7,8)(9,11)(10,17)(12,26)(13,19)(14,23)(15,33)(16,24)(18,35)(20,31)(21,29)(22,32)(25,34)(27,36)(28,38)(30,39)(37,40);(1,3,5,4)(2,7,6,8)(9,12,15,29,16,30,14,28)(10,18,19,25,20,36,22,37)(11,21,23,26,24,38,33,39)(13,27,17,34,32,35,31,40);(1,4,5,3)(2,8,6,7)(9,13,14,31,16,32,15,17)(10,11,22,33,20,24,19,23)(12,27,28,40,30,35,29,34)(18,21,37,39,36,38,25,26);(2,6)(7,8)(9,14,16,15)(10,19,20,22)(11,23,24,33)(12,28,30,29)(13,31,32,17)(18,25,36,37)(21,26,38,39)(27,40,35,34)Polycyclic presentation:
2^9|c^2=fg,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, 7468966);