Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994043 |
| 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 | #2856 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1511 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(12,20)(15,21)(19,24)(22,23)(26,30)(28,31)(32,38)(33,34)(35,40)(37,39);(2,5)(3,7)(9,10,13,16)(11,17,18,14)(12,21,20,15)(19,22,24,23)(25,26,31,33,29,34,28,30)(27,35,39,38,36,32,37,40);(2,5)(6,8)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,24)(21,23)(25,27)(26,32)(28,37)(29,36)(30,38)(31,39)(33,40)(34,35);(1,3,4,7)(2,6,5,8)(11,18)(14,17)(19,24)(22,23)(26,33)(27,36)(28,31)(30,34)(32,38)(35,40);(9,12,13,20)(10,15,16,21)(11,19,18,24)(14,22,17,23)Polycyclic presentation:
2^9|b^2=fgh,d^2=g,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,f^d=fi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72282784, #72283036, #72283256, … |
| maximal normal subgroup of | #72282784, #72283036, #72283256, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7095811);