Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
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 | |
1814 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,9,8)(7,11,12,10)(13,14,17,16)(15,18,19,20)(22,26)(23,29)(24,32)(34,36);(6,8)(10,11)(13,15)(14,18)(16,20)(17,19)(21,22,25,26)(23,30,29,27)(24,33,32,28)(31,34,35,36);(1,2,3,4)(7,12)(10,11)(13,16,17,14)(15,18,19,20)(21,22)(23,30)(24,28)(25,26)(27,29)(31,36)(32,33)(34,35);(5,7,9,12)(6,10,8,11)(21,23)(22,27)(24,31)(25,29)(26,30)(28,34)(32,35)(33,36);(5,8,9,6)(7,11,12,10)(21,24,25,32)(22,28,26,33)(23,31,29,35)(27,34,30,36)Polycyclic presentation:
2^9|a^2=fh,b^2=g,c^2=i,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 483290);