Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C25 |
| commutator | C22 | C24×C4 |
| prime radical | C2×H(3,𝔽4) | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1244 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,4,3)(5,6)(7,11)(8,9)(10,12)(13,14,18,19)(15,17,16,20);(6,9)(10,11)(14,19)(15,16);(1,2,4,3)(13,15,18,16)(14,20,19,17);(5,6)(7,10)(8,9)(11,12)(13,16,18,15)(14,17,19,20);(1,3,4,2)(5,7,8,12)(6,10,9,11)(13,17)(14,15)(16,19)(18,20)Polycyclic presentation:
2^8|a^2=h,c^2=h,d^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df76 conjugacy classes — view the full character table.
| Fitting subgroup of | #34529⋊C3 |
| maximal subgroup of | #97008, #209150, #209151, … |
| maximal normal subgroup of | #97008, #209150, #209151, … |
| maximal quotient of | #513880, #513964, #513972, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27517);