Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| automorphism | Aut(G) order 8388608 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2034 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,9,8)(7,10,12,11)(14,15)(16,20)(21,22)(23,27)(24,26)(25,32)(28,33)(29,30)(31,36)(34,35);(6,8)(10,11)(13,14,17,15)(16,19,20,18)(22,24)(23,29)(28,32)(31,35);(5,7,9,12)(6,10,8,11)(13,15,17,14)(16,18,20,19)(21,23,26,29)(22,27,24,30)(25,31,33,35)(28,34,32,36);(1,2)(3,4)(5,8,9,6)(7,11,12,10)(13,16,17,20)(14,18,15,19)(21,24,26,22)(23,30,29,27)(25,32,33,28)(31,36,35,34);(21,25,26,33)(22,28,24,32)(23,31,29,35)(27,34,30,36)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=f,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4836131);