Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1896 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,8,10)(7,9,11,12)(13,14,16,18)(15,19,20,17)(21,22,26,23)(24,27,31,29)(25,28,34,30)(32,35,33,36);(1,2,3,4)(15,20)(17,19)(22,23)(24,31)(28,30)(32,33);(1,2)(3,4)(13,15)(14,17)(16,20)(18,19)(21,23,26,22)(24,29,31,27)(25,28,34,30)(32,35,33,36);(5,6)(7,9)(8,10)(11,12)(13,15)(14,17)(16,20)(18,19)(21,24)(22,27)(23,29)(25,33)(26,31)(28,36)(30,35)(32,34);(5,7,8,11)(6,9,10,12)(13,14)(15,17)(16,18)(19,20)(21,25)(22,28)(23,30)(24,32)(26,34)(27,35)(29,36)(31,33)Polycyclic presentation:
2^9|a^2=fi,b^2=fgh,c^2=f,e^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3538480);