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 | |
1984 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)(15,19)(16,17)(18,20)(21,22,27,25)(23,30,28,24)(26,29,35,33)(31,36,34,32);(2,4)(14,16)(18,19)(21,23,27,28)(22,24,25,30)(26,34,35,31)(29,36,33,32);(1,2,3,4)(15,20)(18,19)(21,24,27,30)(22,28,25,23)(26,32,35,36)(29,34,33,31);(5,6)(7,9)(8,10)(11,12)(13,15)(14,18)(16,19)(17,20)(21,25,27,22)(23,24,28,30)(26,29,35,33)(31,36,34,32);(5,7,8,11)(6,9,10,12)(13,16,17,14)(15,18,20,19)(21,26)(22,29)(23,31)(24,32)(25,33)(27,35)(28,34)(30,36)Polycyclic presentation:
2^9|a^2=fi,b^2=g,c^2=h,d^2=g,e^2=i,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, 3127306);