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) = 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 | |
417 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(5,6)(7,9)(8,10)(11,12)(13,14,16,18)(15,21,23,19)(17,20,24,26)(22,27,25,28);(1,2,4,3)(5,7,8,11)(6,9,10,12)(14,18)(15,22)(17,24)(19,27)(21,28)(23,25);(6,10)(9,12)(13,15,16,23)(14,19,18,21)(17,25,24,22)(20,27,26,28);(1,3,4,2)(14,18)(15,23)(20,26)(22,25)Polycyclic presentation:
2^8|a^2=h,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×#25029, #6443998, #6444006, … |
| maximal normal subgroup of | C2×#25029, #6443998, #6444006, … |
| maximal quotient of | #281187, #281188, #281189, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18017);