Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394038 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2290 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,15,20)(11,16,23,32)(12,24,26,35)(13,19,14,18)(17,28,34,27)(21,33,37,40)(22,36,29,31)(25,30,38,39);(1,3,4,7)(2,5,6,8)(9,11)(10,16)(12,25)(13,22)(14,29)(15,23)(17,33)(18,31)(19,36)(20,32)(21,28)(24,39)(26,38)(27,37)(30,35)(34,40);(9,12)(10,17)(11,21)(13,28)(14,27)(15,26)(16,30)(18,35)(19,24)(20,34)(22,38)(23,37)(25,29)(31,40)(32,39)(33,36);(2,6)(5,8)Polycyclic presentation:
2^9|a^2=i,b^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,g^b=gi,e^c=ei,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312299);