Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1339 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #327 |
| commutator | C2×C8 | C4×C8 |
| Frattini, cosocle | C22×#97 | C22 |
| socle | C22 | #1294 |
| prime radical | C24 | #103 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
356 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10,13,19,16,22,15,21)(11,23,25,39,27,38,28,35)(12,29,31,18,32,20,14,33)(17,26,34,30,37,40,36,24);(2,5,7,6)(9,11,14,26,15,28,32,24,16,27,31,40,13,25,12,30)(10,17,20,35,19,34,33,23,22,37,29,39,21,36,18,38)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fi,d^2=i,e^2=h,f^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fi,h^a=hi,c^b=cg,g^b=gi,e^c=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#93395, #72213051, #72213052, … |
| maximal normal subgroup of | C2×#93395, #72213051, #72213052, … |
| maximal quotient of | #72211432, C2×#93395 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 591);