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 | |
1854 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(6,8)(10,12)(14,18)(16,19)(22,23)(24,31)(25,33)(34,35);(7,11)(10,12)(14,18)(15,20)(21,22,26,23)(24,27,31,29)(25,30,33,28)(32,35,36,34);(5,6,9,8)(7,10,11,12)(13,14,17,18)(15,19,20,16)(21,23,26,22)(24,29,31,27)(25,30,33,28)(32,35,36,34);(5,7,9,11)(6,10,8,12)(13,15)(14,19)(16,18)(17,20)(21,24)(22,27)(23,29)(25,32)(26,31)(28,34)(30,35)(33,36);(5,8,9,6)(7,12,11,10)(13,16,17,19)(14,15,18,20)(21,25)(22,28)(23,30)(24,32)(26,33)(27,34)(29,35)(31,36)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=fgh,e^2=fg,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, 464427);