Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9938 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1439 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
883 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(5,7)(9,10,14,18,15,20,13,19)(11,16,22,32,24,33,23,31)(12,25,28,17,27,36,29,35)(21,34,26,39,37,40,38,30);(1,2)(3,7,8,5)(4,6)(9,11)(10,16)(12,26,27,38)(13,23)(14,22)(15,24)(17,34,35,40)(18,32)(19,31)(20,33)(21,28,37,29)(25,30,36,39);(1,3)(2,5)(4,8)(6,7)(9,12)(10,17)(11,21)(13,29)(14,28)(15,27)(16,30)(18,36)(19,25)(20,35)(22,26)(23,38)(24,37)(31,40)(32,34)(33,39)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=cf,e^b=eh,f^b=fhi,d^c=dh,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 23464);