Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30484 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2361 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
997 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)(10,16,20,19)(12,25,27,28)(17,21,32,34)(22,36,38,37);(1,3,4,7)(2,5,6,8)(9,10)(11,21)(12,26)(13,19)(14,16)(15,20)(17,23)(18,25)(22,31)(24,34)(27,30)(28,33)(29,32)(35,36)(37,39)(38,40);(2,6)(5,8)(9,11,13,23,15,24,14,29)(10,17,19,34,20,32,16,21)(12,22,28,37,27,38,25,36)(18,31,26,39,33,40,30,35);(2,6)(5,8)(9,12)(10,18)(11,22)(13,28)(14,25)(15,27)(16,30)(17,31)(19,26)(20,33)(21,35)(23,37)(24,38)(29,36)(32,40)(34,39)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=eh,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,e^d=ei,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296005);