Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 2097152 |
| 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 | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2610 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(6,9)(7,11)(14,18)(16,20)(21,22,25,29)(23,27,31,26)(24,28,34,33)(30,36,32,35);(2,4)(6,9)(10,12)(14,18)(15,19)(21,23,25,31)(22,26,29,27)(24,30,34,32)(28,35,33,36);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,17,18)(15,16,19,20)(21,23)(22,27)(24,32)(25,31)(26,29)(28,35)(30,34)(33,36);(5,6)(7,10)(8,9)(11,12)(13,15,17,19)(14,16,18,20)(21,22)(23,27)(24,33)(25,29)(26,31)(28,34)(30,35)(32,36);(5,7)(6,10)(8,11)(9,12)(13,16)(14,19)(15,18)(17,20)(21,24)(22,28)(23,30)(25,34)(26,35)(27,36)(29,33)(31,32)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=fgi,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, 3437236);