Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519083 |
| 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 |
| prime radical | #61096 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1893 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,6)(3,7,4,8)(9,10)(11,16)(12,13)(14,19)(15,21)(17,29)(18,31)(20,28)(22,35)(23,27)(24,33)(25,37)(26,34)(30,32)(36,38)(39,40);(2,6)(7,8)(9,11,15,26)(10,16,21,34)(12,27,30,40)(13,23,32,39)(14,24,28,25)(17,35,31,38)(18,36,29,22)(19,33,20,37);(1,3,5,4)(2,7,6,8)(9,12,28,18,15,30,14,29)(10,17,19,32,21,31,20,13)(11,22,24,40,26,36,25,27)(16,23,37,38,34,39,33,35);(1,3,5,4)(2,7,6,8)(9,13,28,17,15,32,14,31)(10,18,19,30,21,29,20,12)(11,23,24,38,26,39,25,35)(16,22,37,40,34,36,33,27);(1,4,5,3)(2,7,6,8)(9,14,15,28)(10,19,21,20)(11,24,26,25)(12,18,30,29)(13,31,32,17)(16,33,34,37)(22,27,36,40)(23,38,39,35)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fgi,d^2=fgi,e^2=fi,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,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7473253);