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 | |
1960 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)(7,9)(8,10)(11,12)(13,14)(15,20)(16,18)(17,19)(21,22)(23,30)(24,26)(25,33)(27,29)(28,35)(31,36)(32,34);(6,7)(8,12)(13,15)(14,17)(16,20)(18,19)(21,23,27,24)(22,26,29,30)(25,31,35,32)(28,36,33,34);(6,7)(10,11)(14,16)(15,19)(21,24,27,23)(22,26,29,30)(25,31,35,32)(28,34,33,36);(1,2,3,4)(5,7,9,6)(8,10,12,11)(13,16,18,14)(15,17,19,20)(21,25)(22,28)(23,31)(24,32)(26,36)(27,35)(29,33)(30,34);(5,8,9,12)(6,10,7,11)(13,17)(14,19)(15,16)(18,20)(21,26)(22,23)(24,29)(25,34)(27,30)(28,32)(31,33)(35,36)Polycyclic presentation:
2^9|b^2=fg,c^2=fg,d^2=fgi,e^2=gh,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, 3047447);