Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417581 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#447 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(9,10)(11,12)(13,14)(15,21)(16,25)(17,19)(18,23)(20,22)(24,27)(26,28);(5,6)(7,8)(13,15,22,28,17,24,16,23)(14,18,25,27,19,26,20,21);(2,4)(6,8)(14,19)(18,26)(20,25)(21,27);(10,12)(15,23)(16,22)(18,21)(20,25)(24,28)(26,27)Polycyclic presentation:
2^9|a^2=fi,b^2=ei,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=dei,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72311823, #72311837, #72311863, … |
| maximal normal subgroup of | #72311823, #72311837, #72311863, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326142);