Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514148 |
| 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 | |
1763 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)(3,8)(4,6)(5,7)(9,10)(11,16)(12,25)(13,31)(14,19)(15,21)(17,29)(18,24)(20,32)(22,35)(23,34)(26,33)(27,36)(28,37)(30,38)(39,40);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,26)(13,32)(14,23)(15,24)(17,35)(18,21)(19,34)(20,31)(22,29)(25,33)(27,30)(28,39)(36,38)(37,40);(9,12,15,30)(10,17,21,37)(11,22,24,39)(13,27,23,26)(14,28,32,29)(16,33,18,36)(19,25,20,38)(31,35,34,40);(1,4,5,3)(2,6,7,8)(9,13)(10,18)(11,14)(12,27)(15,23)(16,21)(17,36)(19,31)(20,34)(22,28)(24,32)(25,35)(26,30)(29,39)(33,37)(38,40);(2,7)(6,8)(9,14,15,32)(10,19,21,20)(11,23,24,13)(12,28,30,29)(16,34,18,31)(17,25,37,38)(22,26,39,27)(33,40,36,35)Polycyclic presentation:
2^9|b^2=f,c^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,g^c=gi,h^c=hi,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7424789);