Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| 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 | |
1020 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)(21,22,26,29,25,28,34,36)(23,30,32,33,31,27,35,24);(2,4)(5,6,9,13,10,14,8,11)(7,12,17,20,15,19,16,18)(21,23)(22,27)(24,29)(25,31)(26,32)(28,30)(33,36)(34,35);(5,7)(6,12)(8,16)(9,17)(10,15)(11,18)(13,20)(14,19)(21,24,26,30,25,33,34,27)(22,23,29,32,28,31,36,35);(7,15)(12,19)(16,17)(18,20)(22,28)(23,31)(29,36)(32,35)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=di,d^c=dh128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 89736);