Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | Aut(G) order 65536 |
| 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 | C2×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1257 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(14,15)(19,20);(5,6,8,10)(7,9,12,11)(13,14,18,15)(16,19,17,20);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,15,18,14)(16,20,17,19);(5,7)(6,9)(8,12)(10,11)(13,16,18,17)(14,19,15,20);(5,6)(7,11)(8,10)(9,12)(13,17,18,16)(14,20,15,19)Polycyclic presentation:
2^8|b^2=fh,c^2=f,d^2=f,e^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 | #6576544, #6576545, #6576568, … |
| maximal normal subgroup of | #6576544, #6576545, #6576568, … |
| maximal quotient of | #519005, #519006, #519031, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29222);