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 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
956 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)(11,17)(12,15)(13,22)(14,20)(16,21)(18,23)(19,24);(1,3)(2,5)(4,8)(6,7)(9,10,12,15)(11,18,19,14)(13,16,21,22)(17,20,24,23);(9,11,13,20,12,19,21,23)(10,14,16,17,15,18,22,24);(3,8)(5,7)(9,10)(11,18)(12,15)(13,16)(14,19)(17,23)(20,24)(21,22);(2,6)(5,7)(10,15)(14,18)(16,22)(17,24)Polycyclic presentation:
2^8|b^2=h,c^2=g,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 | #130446, #130448, #131737, … |
| maximal normal subgroup of | #130446, #130448, #131737, … |
| maximal quotient of | #6386769, #6389182, #6389325, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55328);