Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1014 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(6,9)(7,11)(13,14)(15,19)(16,18)(17,20)(21,22)(23,31)(24,33)(25,30)(26,27)(28,34)(29,32)(35,36);(5,6,8,9)(7,12,11,10)(13,15)(14,17)(16,20)(18,19)(21,23,25,26)(22,27,30,31)(24,34,35,32)(28,33,29,36);(14,18)(15,20)(21,24,26,34,25,35,23,32)(22,28,31,33,30,29,27,36);(5,7,8,11)(6,10,9,12)(15,20)(17,19)(21,24,23,32,25,35,26,34)(22,29,27,33,30,28,31,36)Polycyclic presentation:
2^9|a^2=h,b^2=egi,c^2=fgi,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 294535);