Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5024635 |
| 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 | |
2129 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)(7,9)(11,12)(15,19)(17,20)(21,22,26,27)(23,24,29,32)(25,33,36,28)(30,35,34,31);(5,6,10,8)(7,12,9,11)(13,14,16,18)(15,17,19,20)(22,27)(23,29)(28,33)(30,34);(5,7,10,9)(6,11,8,12)(13,14)(15,17)(16,18)(19,20)(21,23)(22,24)(25,34)(26,29)(27,32)(28,35)(30,36)(31,33);(1,2)(3,4)(5,8,10,6)(7,11,9,12)(21,24)(22,23)(25,35)(26,32)(27,29)(28,34)(30,33)(31,36);(5,9,10,7)(6,11,8,12)(13,15,16,19)(14,17,18,20)(21,25)(22,28)(23,30)(24,31)(26,36)(27,33)(29,34)(32,35)Polycyclic presentation:
2^9|a^2=f,b^2=fh,c^2=g,d^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=dg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4932969);