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 |
| prime radical | #23950 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1723 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)(12,24)(13,19)(14,28)(15,20)(17,26)(18,22)(21,34)(23,33)(25,31)(27,35)(29,32)(30,36)(37,39)(38,40);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,25)(13,22)(14,29)(15,23)(17,34)(18,19)(20,33)(21,26)(24,31)(27,37)(28,32)(30,38)(35,39)(36,40);(1,4,5,3)(2,6,7,8)(9,12)(10,17)(11,21)(13,26)(14,30)(15,27)(16,31)(18,34)(19,35)(20,36)(22,37)(23,38)(24,28)(25,29)(32,40)(33,39);(9,11)(10,18)(12,21)(13,22)(14,29)(15,23)(16,28)(17,34)(19,33)(20,32)(24,31)(25,30)(26,37)(27,38)(35,39)(36,40);(2,7)(6,8)(9,13,15,14)(10,19,20,28)(11,22,23,29)(12,26,27,30)(16,18,33,32)(17,35,36,24)(21,37,38,25)(31,34,39,40)Polycyclic presentation:
2^9|b^2=f,c^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, 7422633);