Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2190 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1879 |
| prime radical | C24 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
524 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(5,6)(7,14)(8,23)(9,25)(10,16)(11,28)(12,17)(13,19)(15,18)(20,33)(21,34)(22,31)(24,32)(26,30)(27,36)(29,35);(5,7,10,21,12,22,11,29)(6,13,16,32,17,33,28,23)(8,18,19,9,24,26,20,27)(14,30,34,36,31,15,35,25);(1,3,4,2)(5,8)(6,14)(7,18)(9,21)(10,19)(11,20)(12,24)(13,30)(15,33)(16,34)(17,31)(22,26)(23,25)(27,29)(28,35)(32,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=hi,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,d^b=dh,e^b=ehi,h^b=hi,d^c=dhi,e^c=eh,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54454);