Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574077 |
| 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 | #60458 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1983 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)(11,15)(12,17)(13,19)(14,20)(16,29)(18,28)(21,39)(22,33)(23,24)(25,34)(26,35)(27,32)(30,37)(31,38)(36,40);(1,3,5,4)(2,6,7,8)(9,11,14,26)(10,15,20,35)(12,27,31,22)(13,24,28,25)(16,36,37,39)(17,32,38,33)(18,34,19,23)(21,29,40,30);(9,12,28,30,14,31,13,29)(10,16,19,38,20,37,18,17)(11,21,25,22,26,40,24,27)(15,32,23,36,35,33,34,39);(1,4,5,3)(2,6,7,8)(9,12)(10,17)(11,22)(13,29)(14,31)(15,33)(16,19)(18,37)(20,38)(21,24)(23,39)(25,40)(26,27)(28,30)(32,35)(34,36);(9,10)(11,23)(12,16)(13,18)(14,20)(15,24)(17,29)(19,28)(21,36)(22,33)(25,35)(26,34)(27,32)(30,38)(31,37)(39,40)Polycyclic presentation:
2^9|b^2=fi,c^2=gi,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,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, 6491532);