Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
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 | |
744 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)(5,6)(7,8)(9,10)(11,18)(12,16)(13,15)(14,20)(17,22)(19,24)(21,23);(1,2,4,3)(9,11,12,20)(10,14,16,18)(13,19,22,21)(15,23,17,24);(6,8)(10,15)(11,19)(13,22)(16,17)(20,21)(23,24);(1,3,4,2)(10,16)(14,18)(15,17)(23,24)Polycyclic presentation:
2^8|b^2=e,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gh,f^c=fh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24244, #6442628, #6442644, … |
| maximal normal subgroup of | C2×#24244, #6442628, #6442644, … |
| maximal quotient of | #261275, #261287, #261301, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17232);