Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17577 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
498 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,7,6,3,4,8)(9,10,12,16,13,15,21,23)(11,17,18,24,19,14,20,22);(1,3)(2,6)(4,8)(5,7)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22);(1,4,6,5)(2,7,3,8);(10,15)(14,17)(16,23)(22,24)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=h,e^2=h,f^2=gh,b^a=be,c^a=ch,d^a=dg,e^a=eg,e^b=eh,f^b=fgh52 conjugacy classes — view the full character table.
| maximal subgroup of | #105893, #106189, #107409, … |
| maximal normal subgroup of | #105893, #106189, #107409, … |
| maximal quotient of | #160578, #160588, #160603, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10739);