Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500015 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | #62871 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3136 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(9,10)(11,14)(12,23,26,17)(13,18)(15,19)(16,22)(20,25,35,31)(21,30)(24,36,39,32)(27,28)(29,33)(34,40,37,38);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,20)(13,27)(14,19)(15,22)(17,31)(18,30)(21,29)(23,25)(24,37)(26,35)(28,33)(32,40)(34,39)(36,38);(1,3)(2,5)(4,8)(6,7)(9,12)(10,17)(11,20)(13,24)(14,25)(15,26)(16,31)(18,32)(19,23)(21,34)(22,35)(27,37)(28,38)(29,39)(30,40)(33,36);(3,8)(5,7)(9,13,15,29)(10,18,19,33)(11,21,22,27)(12,24,26,39)(14,30,16,28)(17,32,23,36)(20,34,35,37)(25,40,31,38);(2,6)(5,7)(9,14,15,16)(10,11,19,22)(12,25,26,31)(13,28,29,30)(17,20,23,35)(18,27,33,21)(24,38,39,40)(32,37,36,34)Polycyclic presentation:
2^9|a^2=fi,d^2=i,e^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,g^b=gi,d^c=dh,f^c=fi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6407223);