Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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:
(6,9)(10,12)(14,15)(16,20)(21,22,27,30)(23,28,24,25)(26,29,35,36)(31,34,32,33);(2,4)(14,15)(18,19)(21,23,27,24)(22,25,30,28)(26,32,35,31)(29,34,36,33);(1,2,3,4)(13,14,17,15)(16,18,20,19)(21,24,27,23)(22,28,30,25)(26,32,35,31)(29,34,36,33);(5,6,8,9)(7,10,11,12)(13,15,17,14)(16,19,20,18)(21,25,27,28)(22,24,30,23)(26,34,35,33)(29,31,36,32);(5,7,8,11)(6,10,9,12)(13,16)(14,18)(15,19)(17,20)(21,26)(22,29)(23,31)(24,32)(25,33)(27,35)(28,34)(30,36)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=fh,d^2=i,e^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,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, 3178992);