Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
2081 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)(5,6)(7,10)(8,11)(9,12)(13,14)(15,17)(16,18)(19,20)(21,22,26,29)(23,30,28,25)(24,32,35,27)(31,34,33,36);(5,6,9,12)(7,8,11,10)(15,19)(17,20)(23,28)(25,30)(31,33)(34,36);(7,11)(8,10)(13,14,16,18)(15,17,19,20)(21,23,26,28)(22,25,29,30)(24,33,35,31)(27,36,32,34);(1,2,3,4)(5,7,9,11)(6,10,12,8)(13,14)(15,17)(16,18)(19,20)(21,24,26,35)(22,27,29,32)(23,31,28,33)(25,34,30,36);(5,8,9,10)(6,11,12,7)(13,15,16,19)(14,17,18,20)(21,25)(22,28)(23,29)(24,34)(26,30)(27,33)(31,32)(35,36)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=g,d^2=fi,e^2=fgh,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, 4990237);