Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| 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 |
| prime radical | C2×#449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
724 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6)(7,8)(9,10)(11,19)(12,17)(13,18)(14,23)(15,21)(16,20)(22,24);(1,2,3,4)(9,11)(10,15)(12,22)(13,21)(14,20)(16,17)(18,19)(23,24);(9,11)(10,16)(12,20)(13,21)(14,22)(15,17)(18,24)(19,23);(6,8)(9,12,13,14)(10,17,18,23)(11,20,21,22)(15,24,19,16)Polycyclic presentation:
2^8|b^2=eh,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,d^b=dh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #6470292, #6470293, #6470294, … |
| maximal normal subgroup of | #6470292, #6470293, #6470294, … |
| maximal quotient of | #347778, #347781, #347784, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25907);