Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#121 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1016 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(6,11)(8,9)(12,18)(13,14)(16,17)(19,20)(21,22,24,28,25,27,31,34)(23,26,29,33,30,32,35,36);(2,4)(5,6,9,14,10,13,8,11)(7,12,17,19,15,20,16,18);(5,7)(6,12)(8,16)(9,17)(10,15)(11,18)(13,20)(14,19)(21,23,24,29,25,30,31,35)(22,26,28,33,27,32,34,36);(6,13)(7,15)(11,14)(16,17)(22,27)(26,32)(28,34)(33,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,e^2=h,f^2=i,g^2=h,b^a=be,d^a=di,e^a=eh,g^a=gh,d^b=dh,d^c=dh128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 89667);