Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1986 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)(7,11)(8,9)(10,12)(13,14,17,16)(15,18,19,20)(22,23)(24,31)(25,33)(35,36);(5,6)(7,11)(8,9)(10,12)(15,19)(18,20)(21,22,26,23)(24,27,31,29)(25,30,33,28)(32,36,34,35);(6,9)(10,11)(13,15,17,19)(14,18,16,20)(21,23,26,22)(24,27,31,29)(25,28,33,30)(32,36,34,35);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(21,24)(22,27)(23,29)(25,34)(26,31)(28,36)(30,35)(32,33);(5,7)(6,10)(8,12)(9,11)(13,16,17,14)(15,20,19,18)(21,25,26,33)(22,28,23,30)(24,32,31,34)(27,35,29,36)Polycyclic presentation:
2^9|a^2=fgh,b^2=f,c^2=gh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,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, 4059100);