Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23918 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C24 | C4×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
940 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)(5,6)(7,15)(8,11)(9,14)(10,13)(12,17)(16,20)(18,19)(21,22,25,30,27,24,26,29)(23,31,35,32,33,36,34,28);(2,4)(6,11)(8,9)(12,19)(13,14)(15,20)(16,17)(21,23,25,35,27,33,26,34)(22,28,30,31,24,32,29,36);(5,7,10,18)(6,12,13,20)(8,17,9,16)(11,19,14,15)(21,24,27,22)(23,32,33,28)(25,29,26,30)(31,35,36,34);(6,13)(11,14)(12,20)(15,19)(22,24)(23,33)(29,30)(34,35)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fh,d^b=di,e^b=eh,f^b=fh,d^c=di,e^c=eh,f^c=fh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 99196);