Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24398 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#390 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1407 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16,24,34)(13,18)(14,26)(17,19)(20,37,38,29)(21,35)(22,30)(23,32,39,40)(25,36,28,33)(27,31);(2,5)(6,8)(9,11,19,21,13,22,14,27)(10,15,17,35,18,30,26,31)(12,23,25,20,24,39,28,38)(16,32,36,37,34,40,33,29);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,20)(13,24)(14,28)(15,29)(17,36)(18,34)(19,25)(21,39)(22,38)(23,27)(26,33)(30,37)(31,40)(32,35);(2,5)(6,8)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=e,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,g^c=gi,h^c=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273994);