Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2735949 |
| automorphism | Aut(G) order 4194304 |
| 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 | |
1806 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)(6,8)(7,9)(13,14,17,15)(16,18,20,19)(21,22)(23,27)(24,26)(25,28)(29,31)(30,34)(32,33)(35,36);(5,6,10,8)(7,11,9,12)(13,15,17,14)(16,18,20,19)(22,24)(23,29)(28,32)(30,35);(5,7,10,9)(6,11,8,12)(13,16,17,20)(14,18,15,19)(21,23,26,29)(22,27,24,31)(25,30,33,35)(28,34,32,36);(1,2)(3,4)(5,8,10,6)(7,11,9,12)(21,24,26,22)(23,27,29,31)(25,28,33,32)(30,36,35,34);(5,9,10,7)(6,12,8,11)(21,25)(22,28)(23,30)(24,32)(26,33)(27,34)(29,35)(31,36)Polycyclic presentation:
2^9|a^2=fgh,b^2=fh,c^2=gh,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2644420);