Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996335 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #59220 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1580 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26,29,32)(27,33,34,40)(28,36,37,31)(30,38,39,35);(1,2)(3,6)(4,5)(7,8)(11,17)(12,20)(14,21)(15,19);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,27,28,35,29,34,37,38)(26,30,31,33,32,39,36,40);(2,5)(6,8)(11,17)(14,21)(18,23)(22,24)(27,34)(30,39)(33,40)(35,38);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)Polycyclic presentation:
2^9|a^2=hi,c^2=fg,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72282738, #72283173, #72284418 |
| maximal normal subgroup of | #72282738, #72283173, #72284418 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7081360);