Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60750 |
| 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 | C2×#525 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1171 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,17)(12,20)(13,15)(14,18)(16,22)(19,24)(21,23);(1,3)(2,5)(4,8)(6,7);(3,8)(5,7)(9,11,13,19)(10,14,15,23)(12,21,22,18)(16,24,20,17);(2,6)(5,7);(10,15)(14,23)(16,20)(17,24)Polycyclic presentation:
2^8|c^2=h,g^2=h,b^a=bf,c^a=cg,d^a=df,e^a=eh,g^a=gh,c^b=cf,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #114438, #114441, #114450, … |
| maximal normal subgroup of | #114438, #114441, #114450, … |
| maximal quotient of | #6374219, #6374221, #6374872, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53739);