Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993915 |
| 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 | #59057 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1530 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,4)(3,7,8,6)(12,20)(15,21)(19,24)(22,23)(25,26,28,33,29,31,30,34)(27,35,38,32,37,40,36,39);(1,3)(2,6)(4,7)(5,8)(9,10,13,16)(11,17,18,14)(12,21,20,15)(19,22,24,23)(26,31)(32,39)(33,34)(35,40);(1,4,5,2)(3,7,8,6)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,24)(21,23)(25,27,29,37)(26,32,31,39)(28,36,30,38)(33,35,34,40);(3,8)(6,7)(11,18)(14,17)(19,24)(22,23)(25,28,29,30)(26,33,31,34)(27,36,37,38)(32,35,39,40);(9,12,13,20)(10,15,16,21)(11,19,18,24)(14,22,17,23)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=f,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72282648, #72283675, #72283849, … |
| maximal normal subgroup of | #72282648, #72283675, #72283849, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7090077);