Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62253 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
927 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,17)(11,15,13,22)(12,19,21,23)(16,18,24,20);(1,3)(2,5)(4,8)(6,7)(9,11,14,13)(10,15,17,22)(12,20,21,18)(16,23,24,19);(9,12,13,20,14,21,11,18)(10,16,15,19,17,24,22,23);(3,8)(5,7)(12,21)(16,24)(18,20)(19,23);(2,6)(5,7)(9,11,14,13)(10,15,17,22)(12,18,21,20)(16,19,24,23)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=g,e^2=h,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,c^b=ch,d^b=df,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #130443, #130445, #131744, … |
| maximal normal subgroup of | #130443, #130445, #131744, … |
| maximal quotient of | #6386745, #6389129, #6389301, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55252);