Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2366 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1056 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6,10,16,11,15,27,14)(7,17,19,12,20,31,21,30)(8,22,9,25,24,13,26,32)(18,34,35,33,36,29,23,28);(5,7)(6,12,15,30)(8,23)(9,18)(10,19)(11,20)(13,29,22,34)(14,17,16,31)(21,27)(24,35)(25,33,32,28)(26,36);(2,4)(5,8,10,9,11,24,27,26)(6,13,16,32,15,22,14,25)(7,18,21,23,20,36,19,35)(12,28,17,29,30,33,31,34);(5,9,10,24,11,26,27,8)(6,13,14,25,15,22,16,32)(7,18,19,35,20,36,21,23)(12,29,17,33,30,34,31,28)Polycyclic presentation:
2^9|a^2=h,b^2=efh,c^2=g,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275854);