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)(11,17)(12,24)(13,29)(14,20)(15,31)(16,21)(18,27)(19,30)(22,35)(23,34)(25,33)(26,36)(28,37)(32,38)(39,40);(1,3,5,4)(2,6,7,8)(9,11,16,13)(10,17,21,29)(12,25,28,39)(14,23,15,30)(18,35,38,36)(19,20,34,31)(22,32,26,27)(24,33,37,40);(1,4,5,3)(2,6,7,8)(9,12,16,28)(10,18,21,38)(11,22,13,26)(14,27,15,32)(17,33,29,40)(19,36,34,35)(20,37,31,24)(23,39,30,25);(9,13,16,11)(10,19,21,34)(12,26,28,22)(14,30,15,23)(17,31,29,20)(18,36,38,35)(24,40,37,33)(25,32,39,27);(2,7)(6,8)(9,14,16,15)(10,20,21,31)(11,23,13,30)(12,27,28,32)(17,34,29,19)(18,37,38,24)(22,39,26,25)(33,35,40,36)Polycyclic presentation:
2^9|b^2=fi,c^2=fi,d^2=i,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, 7422643);