Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
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 | |
2888 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)(7,11)(10,12)(15,19)(18,20)(22,23)(24,31)(28,30)(32,36);(6,9)(10,12)(14,17)(18,20)(21,22,26,23)(24,27,31,29)(25,30,33,28)(32,35,36,34);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,14)(15,18)(16,17)(19,20)(21,23,26,22)(24,27,31,29)(25,30,33,28)(32,34,36,35);(13,15)(14,18)(16,19)(17,20)(21,24)(22,27)(23,29)(25,32)(26,31)(28,34)(30,35)(33,36);(5,7)(6,10)(8,11)(9,12)(13,15)(14,18)(16,19)(17,20)(21,25,26,33)(22,28,23,30)(24,32,31,36)(27,34,29,35)Polycyclic presentation:
2^9|b^2=f,c^2=hi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
| maximal subgroup of | #72266756, #72266759, #72267630, … |
| maximal normal subgroup of | #72266756, #72266759, #72267630, … |
| maximal quotient of | #72266782, #72267289, #72267675, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4235830);