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 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
478 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)(7,11)(8,10)(9,12)(14,18)(20,26)(21,27)(22,28);(5,7,8,12)(6,9,10,11)(13,14,19,27,16,18,17,21)(15,22,25,20,23,28,24,26);(2,4)(13,15,16,23)(14,20,18,26)(17,25,19,24)(21,28,27,22);(6,10)(9,11)(14,18)(20,26)(21,27)(22,28)Polycyclic presentation:
2^8|b^2=eg,c^2=h,g^2=h,b^a=be,c^a=cf,d^a=deh,c^b=cg,d^b=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#28112, #6459346, #6459353, … |
| maximal normal subgroup of | C2×#28112, #6459346, #6459353, … |
| maximal quotient of | #316631, #316632, #316635, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21100);