Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25440 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C2×D42 |
| prime radical | #60450 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1960 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)(3,7)(4,8)(5,6)(9,10,14,19,15,20,13,17)(11,21,24,32,25,31,22,16)(12,18,28,35,29,36,27,33)(23,34,26,39,38,40,37,30);(2,6)(7,8)(9,11)(10,16)(12,26)(13,24)(14,22)(15,25)(17,21)(18,34)(19,31)(20,32)(23,28)(27,38)(29,37)(30,35)(33,39)(36,40);(1,3,5,4)(2,7,6,8)(10,17,20,19)(11,22,25,24)(18,33,36,35)(23,37,38,26);(1,4,5,3)(2,8,6,7)(9,12,14,28,15,29,13,27)(10,18,19,35,20,36,17,33)(11,23,24,26,25,38,22,37)(16,30,21,34,32,39,31,40);(9,13,15,14)(10,17,20,19)(11,24,25,22)(12,27,29,28)(16,21,32,31)(18,33,36,35)(23,26,38,37)(30,34,39,40)Polycyclic presentation:
2^9|a^2=h,c^2=fg,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,h^b=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6364899);