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 | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
433 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,16,19,17,20,25,28)(15,21,22,27,23,26,24,18);(1,2)(3,4)(13,15,17,23)(14,18,20,27)(16,24,25,22)(19,26,28,21);(2,4)(6,9)(10,12);(5,7,8,11)(6,10,9,12)(14,19)(16,25)(18,26)(20,28)(21,27)(22,24)Polycyclic presentation:
2^8|a^2=e,b^2=h,d^2=f,e^2=h,b^a=be,c^a=cf,d^a=deh,c^b=cg,e^b=eh,e^d=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#26636, #6439528, #6439531, … |
| maximal normal subgroup of | C2×#26636, #6439528, #6439531, … |
| maximal quotient of | #310505, #310507, #310513, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19624);