Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| 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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2101 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,3)(5,6)(7,11)(8,12)(9,10)(15,19)(17,20)(21,22)(23,26)(24,27)(25,30)(28,33)(29,31)(32,36)(34,35);(2,3)(5,7,10,8)(6,11,9,12)(22,23)(24,31)(25,33)(35,36);(1,2,4,3)(5,7,10,8)(6,12,9,11)(13,14,16,18)(15,17,19,20)(21,23,26,22)(24,29,31,27)(25,28,33,30)(32,35,34,36);(5,8,10,7)(6,11,9,12)(13,15)(14,17)(16,19)(18,20)(21,24,26,31)(22,27,23,29)(25,34,33,32)(28,36,30,35);(1,3,4,2)(5,9,10,6)(7,11,8,12)(21,25,26,33)(22,28,23,30)(24,32,31,34)(27,35,29,36)Polycyclic presentation:
2^9|b^2=gh,c^2=hi,d^2=fgh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,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, 3978885);