Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
776 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(3,7)(10,13)(14,16)(17,18)(19,20)(21,22)(23,24);(1,2,5,4)(3,8,7,6)(18,22)(19,23);(1,2,5,4)(3,6,7,8)(9,10,12,13)(11,14,15,16)(17,19,21,23)(18,20,22,24);(1,3,5,7)(2,6,4,8)(17,20)(18,23)(19,22)(21,24);(1,4,5,2)(3,8,7,6)(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^8|b^2=fh,c^2=gh,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #97284, #168043, #168044, … |
| maximal normal subgroup of | #97284, #168043, #168044, … |
| maximal quotient of | #553941, #553942, #554143, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30253);