Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1734 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,4,3)(5,6,9,7)(8,11,12,10)(15,19)(18,20)(21,22,27,29)(23,30,24,26)(25,28,34,36)(31,35,32,33);(6,7)(8,12)(14,17)(18,20)(21,23,27,24)(22,26,29,30)(25,31,34,32)(28,33,36,35);(1,2,4,3)(6,7)(10,11)(13,14,16,17)(15,18,19,20)(21,24,27,23)(22,26,29,30)(25,31,34,32)(28,35,36,33);(5,7,9,6)(8,11,12,10)(13,15,16,19)(14,18,17,20)(21,25)(22,28)(23,31)(24,32)(26,33)(27,34)(29,36)(30,35);(1,3,4,2)(5,8,9,12)(6,10,7,11)(21,26)(22,23)(24,29)(25,33)(27,30)(28,31)(32,36)(34,35)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=ghi,d^2=gh,e^2=fgi,b^a=bf,c^a=cg,d^a=dh,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 508204);