Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31076 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,𝔽4) |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2684 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1031 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(6,12,16,15)(7,17,20,25)(14,21,30,26)(19,33,34,31);(5,6)(7,18)(8,21)(9,15)(10,12)(11,16)(13,17)(14,22)(19,27)(20,28)(23,30)(24,26)(25,29)(31,35)(32,33)(34,36);(5,7,10,25,11,20,9,17)(6,13,12,18,16,29,15,28)(8,19,22,33,24,34,23,31)(14,27,26,32,30,36,21,35);(5,8)(6,14)(7,19)(9,23)(10,22)(11,24)(12,26)(13,27)(15,21)(16,30)(17,31)(18,32)(20,34)(25,33)(28,35)(29,36)Polycyclic presentation:
2^9|a^2=h,c^2=efg,e^2=i,f^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,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293563);