Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #115213 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | #35088 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1817 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,19)(12,23,25,16)(13,26)(14,18)(15,21)(17,28)(20,34,36,40)(22,33)(24,32,39,37)(27,30)(29,38,31,35);(5,8)(9,11)(10,15)(12,20)(13,27)(14,22)(16,31)(17,33)(18,30)(19,26)(21,28)(23,29)(24,38)(25,36)(32,34)(35,39)(37,40);(1,3)(2,5)(4,7)(6,8)(9,12)(10,16)(11,20)(13,24)(14,25)(15,29)(17,32)(18,23)(19,34)(21,35)(22,36)(26,37)(27,38)(28,39)(30,31)(33,40)Polycyclic presentation:
2^9|a^2=g,d^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,d^b=dh,e^b=eh,d^c=di,g^c=gh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72260824, #72260828, #72260849 |
| maximal normal subgroup of | #72260824, #72260828, #72260849 |
| maximal quotient of | #72260935 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22409);