Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24278 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×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 | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1322 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)(12,22,25,16)(13,18)(14,17)(19,35,36,28)(20,30)(21,29)(23,39,37,31)(24,34,26,32)(27,33)(38,40);(2,5)(6,8)(9,11,13,20,14,21,27,38)(10,15,18,30,17,29,33,40)(12,23,24,36,25,37,26,19)(16,31,32,35,22,39,34,28);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,19)(13,26)(14,25)(15,28)(17,22)(18,34)(20,37)(21,36)(23,38)(24,27)(29,35)(30,39)(31,40)(32,33);(2,5)(6,8)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=e,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 269049);