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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2092 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)(8,11)(10,12)(13,14,17,19)(15,16,20,18)(21,22)(23,26)(24,32)(25,29)(27,31)(28,34)(30,36)(33,35);(5,7)(6,8)(9,12)(10,11)(13,15)(14,18)(16,19)(17,20)(21,23,25,31)(22,26,29,27)(24,33,34,30)(28,36,32,35);(5,8,10,9)(6,11,12,7)(13,15,17,20)(14,16,19,18)(21,23)(22,27)(24,30)(25,31)(26,29)(28,36)(32,35)(33,34);(1,2)(3,4)(5,9,10,8)(6,7,12,11)(13,16,17,18)(14,15,19,20)(21,22)(23,26)(24,28)(25,29)(27,31)(30,35)(32,34)(33,36);(5,7)(6,9)(8,12)(10,11)(13,15)(14,16)(17,20)(18,19)(21,24,25,34)(22,28,29,32)(23,30,31,33)(26,35,27,36)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=f,d^2=f,e^2=g,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, 4039589);