Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2314 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)(5,6,9,7)(8,10,12,11)(13,14,17,18)(15,16,19,20)(22,26)(23,29)(24,31)(35,36);(6,7)(10,11)(14,18)(16,20)(21,22,25,26)(23,27,29,34)(24,28,31,32)(30,35,33,36);(5,6,9,7)(8,11,12,10)(13,15,17,19)(14,16,18,20)(21,22)(23,27)(24,32)(25,26)(28,31)(29,34)(30,36)(33,35);(1,2,3,4)(5,7,9,6)(8,10,12,11)(13,16,17,20)(14,15,18,19)(21,23)(22,27)(24,33)(25,29)(26,34)(28,36)(30,31)(32,35);(5,8)(6,10)(7,11)(9,12)(13,15)(14,16)(17,19)(18,20)(21,24)(22,28)(23,30)(25,31)(26,32)(27,35)(29,33)(34,36)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=fg,d^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,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, 3375438);