Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| 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 |
| prime radical | C2×#496 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
524 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)(5,6,8,7)(9,10)(11,17)(12,14)(13,16)(15,19)(18,23)(20,22)(21,24);(1,2,4,3)(5,7,8,6)(10,14)(11,18)(13,20)(15,23)(17,24)(19,21);(6,7)(9,11,12,19)(10,15,14,17)(13,21,20,18)(16,23,22,24);(1,3,4,2)(10,14)(11,19)(16,22)(18,21)Polycyclic presentation:
2^8|a^2=fh,b^2=ef,c^2=h,d^2=eh,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,d^c=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25383, #6444152, #6444162, … |
| maximal normal subgroup of | C2×#25383, #6444152, #6444162, … |
| maximal quotient of | #287875, #287876, #287877, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18371);