Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62265 |
| 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 |
| prime radical | #3548 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
910 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,15,19)(11,21,22,17)(12,14,20,16)(13,18,24,23);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,23)(13,16)(14,24)(15,22)(18,20)(19,21);(9,12,16,10,15,20,14,19)(11,18,13,21,22,23,24,17);(3,8)(5,7)(9,13)(10,18)(11,14)(12,17)(15,24)(16,22)(19,23)(20,21);(2,6)(5,7)(9,14,15,16)(10,12,19,20)(11,13,22,24)(17,18,21,23)Polycyclic presentation:
2^8|a^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,e^b=eh,d^c=dh,e^d=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #263426, #263430, #263454, … |
| maximal normal subgroup of | #263426, #263430, #263454, … |
| maximal quotient of | #6386770, #6389185, #6389324, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55331);