Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573635 |
| 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 | #60375 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2557 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)(3,8)(4,6)(5,7)(9,10)(11,23)(12,26)(13,19)(14,20)(15,21)(16,24)(17,22)(18,28)(25,36)(27,40)(29,34)(30,37)(31,39)(32,38)(33,35);(1,3,5,4)(2,6,7,8)(9,11)(10,12)(13,17)(14,29)(15,26)(16,18)(19,24)(20,35)(21,23)(22,28)(25,31)(27,39)(30,33)(32,40)(34,37)(36,38);(9,12,15,23)(10,18,21,13)(11,24,26,22)(14,27,30,36)(16,19,17,28)(20,33,37,29)(25,39,40,38)(31,34,32,35);(1,4,5,3)(2,6,7,8)(9,13,15,18)(10,19,21,28)(11,17,26,16)(12,24,23,22)(14,25,30,40)(20,36,37,27)(29,31,33,32)(34,39,35,38);(9,14,15,30)(10,20,21,37)(11,25,26,40)(12,27,23,36)(13,29,18,33)(16,31,17,32)(19,34,28,35)(22,38,24,39)Polycyclic presentation:
2^9|b^2=fi,c^2=i,d^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,g^c=gi,h^c=hi,e^d=ehi,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6484155);