Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483928 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #33520 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
725 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(10,16)(11,20,23,40)(12,15)(14,27)(17,32,31,33)(19,36)(21,37,34,38)(22,29,30,39)(25,26)(28,35);(1,3)(2,5)(4,8)(6,7)(9,10)(11,21,23,34)(12,24)(13,16)(14,28)(15,18)(17,30,31,22)(19,26)(20,38,40,37)(25,35)(27,36)(29,33,39,32);(3,8)(5,7)(9,11)(10,17)(12,22)(13,23)(14,29)(15,30)(16,31)(18,21)(19,37)(20,25)(24,34)(26,39)(27,40)(28,38)(32,35)(33,36);(2,6)(5,7)(9,12,13,15)(10,18,16,24)(11,22,23,30)(14,25,26,27)(17,21,31,34)(19,35,28,36)(20,39,40,29)(32,38,33,37)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ceh,d^b=di,f^b=fi,f^c=fhi,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72264749, #72264753, #72264825, … |
| maximal normal subgroup of | #72264749, #72264753, #72264825, … |
| maximal quotient of | #72320463, C2×#496835, #72343723, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404031);