Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | Aut(G) order 524288 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1221 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)(5,6)(7,10)(8,11)(9,12)(13,14,17,19)(15,16,18,20);(5,7,8,12)(6,9,11,10)(13,15)(14,16)(17,18)(19,20);(1,2,3,4)(5,6,8,11)(7,10,12,9)(13,16)(14,18)(15,19)(17,20);(5,6)(7,9)(8,11)(10,12)(13,15)(14,16)(17,18)(19,20);(5,7)(6,10)(8,12)(9,11)(13,16)(14,18)(15,19)(17,20)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=g,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 | #98232, #98238, #130629, … |
| maximal normal subgroup of | #98232, #98238, #130629, … |
| maximal quotient of | #519057, #519093, #519393, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29168);