Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17597 |
| automorphism | Aut(G) order 8192 |
| 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 | C2×#41 | D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
454 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,7,6,3,4,8)(10,14)(11,17)(12,19)(15,22)(16,20)(21,23);(1,3)(2,6)(4,8)(5,7)(9,10,13,16)(11,15,17,23)(12,20,19,14)(18,22,24,21);(1,4,6,5)(2,7,3,8)(9,11,13,17)(10,15,16,23)(12,18,19,24)(14,21,20,22);(11,17)(15,23)(18,24)(21,22)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,e^a=eg,e^b=eh,f^b=fgh,d^c=dg49 conjugacy classes — view the full character table.
| maximal subgroup of | #221272, #221273, #221282, … |
| maximal normal subgroup of | #221272, #221273, #221282, … |
| maximal quotient of | #160643, #160651, #160652, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10751);