Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573643 |
| 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 | #60376 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2599 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)(11,22)(12,29)(13,18)(14,19)(15,21)(16,26)(17,32)(20,33)(23,39)(24,30)(25,28)(27,36)(31,34)(35,38)(37,40);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,30)(13,24)(14,26)(15,27)(17,37)(18,35)(19,22)(20,36)(21,25)(23,32)(28,33)(29,38)(31,39)(34,40);(9,12)(10,17)(11,23)(13,20)(14,18)(15,34)(16,24)(19,31)(21,29)(22,38)(25,37)(26,40)(27,30)(28,35)(32,33)(36,39);(1,4,5,3)(2,6,7,8)(9,13)(10,18)(11,24)(12,19)(14,29)(15,17)(16,35)(20,34)(21,32)(22,30)(23,25)(26,38)(27,37)(28,39)(31,33)(36,40);(9,10)(11,25)(12,31)(13,32)(14,19)(15,20)(16,27)(17,18)(21,33)(22,28)(23,39)(24,35)(26,36)(29,34)(30,38)(37,40)Polycyclic presentation:
2^9|b^2=fi,d^2=fi,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,e^c=ei,g^c=gi,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6483948);