Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2123 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(6,9)(7,11)(13,14)(15,18)(16,20)(17,19)(21,22,25,26)(23,27,31,29)(24,28,33,34)(30,35,32,36);(5,6,8,9)(7,10,11,12)(13,15)(14,18)(16,19)(17,20)(22,26)(27,29)(28,34)(35,36);(6,9)(7,11)(14,19)(15,20)(21,23,25,31)(22,27,26,29)(24,32,33,30)(28,36,34,35);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,20)(18,19)(21,22)(23,29)(24,28)(25,26)(27,31)(30,36)(32,35)(33,34);(5,7,8,11)(6,10,9,12)(13,16)(14,15)(17,18)(19,20)(21,24,25,33)(22,28,26,34)(23,30,31,32)(27,35,29,36)Polycyclic presentation:
2^9|a^2=fgh,b^2=gh,c^2=fgh,d^2=fgi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4989880);