Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2256 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,3)(6,10)(8,11)(13,14,16,18)(15,19,20,17)(21,22)(23,26)(24,29)(25,30)(27,32)(28,33)(31,34)(35,36);(15,20)(17,19)(22,23)(25,33)(27,29)(31,36);(1,2,4,3)(5,6)(7,11)(8,12)(9,10)(13,15)(14,17)(16,20)(18,19)(21,23,26,22)(24,27,32,29)(25,30,33,28)(31,34,36,35);(1,2,4,3)(5,7,9,12)(6,11,10,8)(21,24,26,32)(22,27,23,29)(25,31,33,36)(28,34,30,35);(1,3,4,2)(5,8,9,11)(6,12,10,7)(13,14)(15,17)(16,18)(19,20)(21,25)(22,28)(23,30)(24,31)(26,33)(27,34)(29,35)(32,36)Polycyclic presentation:
2^9|a^2=fh,c^2=i,d^2=fgh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1070022);