Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25140 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3055 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
388 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)(3,6)(4,5)(7,8)(9,10,14,16,12,17,13,18)(11,19,22,23,20,15,21,24);(2,5)(6,8)(9,11,13,21,12,20,14,22)(10,15,18,23,17,19,16,24);(1,3,4,7)(2,6,5,8)(10,16)(11,20)(13,14)(15,23)(17,18)(19,24);(2,5)(6,8)(10,17)(15,19)(16,18)(23,24)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,f^c=fh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25270, #6441696, #6441697, … |
| maximal normal subgroup of | C2×#25270, #6441696, #6441697, … |
| maximal quotient of | #285663, #285666, #285667, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18258);