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 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
561 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)(3,4)(9,10,13,18,15,20,14,19)(11,16,21,17,12,23,22,24);(2,4)(5,6)(7,8)(9,11,13,21,15,12,14,22)(10,16,18,17,20,23,19,24);(6,8)(9,12,15,11)(10,17,20,24)(13,21,14,22)(16,19,23,18);(2,4)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=h,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×#25110, #6438915, #6438944, … |
| maximal normal subgroup of | C2×#25110, #6438915, #6438944, … |
| maximal quotient of | #281535, #281537, #281543, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18098);