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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1755 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)(3,6)(4,5)(7,8)(9,10)(11,20)(12,24)(13,27)(14,31)(15,19)(16,23)(17,29)(18,25)(21,36)(22,33)(26,35)(28,38)(30,40)(32,37)(34,39);(2,5)(6,8)(9,11)(10,16)(12,25)(13,28)(14,22)(15,23)(17,32)(18,33)(19,20)(21,30)(24,31)(26,29)(27,36)(34,35)(37,39)(38,40);(1,3,4,7)(2,6,5,8)(9,12,25,11,15,22,14,23)(10,16,31,33,19,20,18,24)(13,26,39,21,30,37,29,28)(17,32,40,36,34,35,27,38);(9,13,14,29,15,30,25,39)(10,17,18,27,19,34,31,40)(11,21,12,26,23,28,22,37)(16,32,24,38,20,35,33,36);(2,5)(6,8)(9,14,15,25)(10,18,19,31)(11,22,23,12)(13,29,30,39)(16,33,20,24)(17,27,34,40)(21,37,28,26)(32,36,35,38)Polycyclic presentation:
2^9|c^2=fg,d^2=g,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,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7467892);