Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
558 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(9,10,13,17,12,16,19,15)(11,18,22,24,21,14,20,23);(1,2,4,3)(5,6)(7,8)(9,11,19,20,12,21,13,22)(10,14,15,24,16,18,17,23);(6,8)(10,15)(11,20)(13,19)(14,18)(16,17)(21,22);(1,3,4,2)(10,16)(14,18)(15,17)(23,24)Polycyclic presentation:
2^8|a^2=f,b^2=ef,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25388, #6444337, #6444354, … |
| maximal normal subgroup of | C2×#25388, #6444337, #6444354, … |
| maximal quotient of | #287743, #287749, #287755, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18376);