Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| 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 | |
1874 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)(7,9)(8,10)(11,12)(13,14)(15,19)(16,17)(18,20)(22,26)(23,29)(24,31)(34,36);(6,7)(8,12)(14,17)(15,20)(21,22,25,26)(23,27,29,33)(24,32,31,28)(30,36,35,34);(1,3,4,2)(13,14)(15,18)(16,17)(19,20)(21,22)(23,27)(24,28)(25,26)(29,33)(30,34)(31,32)(35,36);(5,7,9,6)(8,11,12,10)(13,15)(14,18)(16,20)(17,19)(21,23)(22,27)(24,30)(25,29)(26,33)(28,34)(31,35)(32,36);(5,8)(6,10)(7,11)(9,12)(13,14)(15,18)(16,17)(19,20)(21,24,25,31)(22,28,26,32)(23,30,29,35)(27,34,33,36)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=i,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dfg,e^b=ef104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 464718);