Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | #2430 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(13,14,18,17)(15,20,16,19);(5,6,8,10)(7,9,12,11)(13,15,18,16)(14,19,17,20);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,16,18,15)(14,20,17,19);(5,7)(6,9)(8,12)(10,11)(13,17,18,14)(15,20,16,19);(5,6)(7,11)(8,10)(9,12)Polycyclic presentation:
2^8|a^2=f,b^2=fh,c^2=f,d^2=f,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh,e^d=eh70 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#36227, #7629883, #7630042, … |
| maximal normal subgroup of | C2×#36227, #7629883, #7630042, … |
| maximal quotient of | #519015, #519016, #519037, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29215);