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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6)(7,10)(8,12)(9,11)(13,14,17,20)(15,16,19,18)(21,22)(23,27)(24,28)(25,26)(29,34)(30,35)(31,33)(32,36);(2,4)(5,7,9,12)(6,10,11,8)(15,19)(16,18)(22,26)(23,29)(28,31)(30,32);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,15,17,19)(14,18,20,16)(21,22)(23,27)(24,31)(25,26)(28,33)(29,34)(30,36)(32,35);(5,7)(6,8)(9,12)(10,11)(13,16,17,18)(14,19,20,15)(21,23,25,29)(22,27,26,34)(24,32,33,30)(28,36,31,35);(5,8)(6,7)(9,10)(11,12)(13,14)(15,18)(16,19)(17,20)(21,24)(22,28)(23,30)(25,33)(26,31)(27,35)(29,32)(34,36)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=gi,d^2=fgi,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, 3486785);