Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
401 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,9,12)(7,11,8,10)(13,14,17,22)(15,20,24,19)(16,25,18,21)(23,27,26,28);(5,7,9,8)(6,10,12,11)(13,15,16,26,17,24,18,23)(14,19,25,27,22,20,21,28);(2,4)(13,15,17,24)(14,20,22,19)(16,23,18,26)(21,28,25,27);(5,8,9,7)(6,11,12,10)(13,16,17,18)(14,21,22,25)(15,23,24,26)(19,27,20,28)Polycyclic presentation:
2^8|a^2=e,b^2=eg,c^2=h,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#28277, #6465997, #6466005, … |
| maximal normal subgroup of | C2×#28277, #6465997, #6466005, … |
| maximal quotient of | #318885, #318886, #318889, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21265);