Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| 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 ◁ #529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
951 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(3,7)(9,10,12,16)(11,18,19,14)(13,15,21,17)(20,23,24,22);(1,2)(3,6)(4,5)(7,8)(9,11,12,19)(10,14,16,18)(13,20,21,24)(15,22,17,23);(10,15)(13,21)(14,22)(16,17)(18,23)(20,24);(2,5)(6,8)(11,19)(14,18)(20,24)(22,23);(1,3,4,7)(2,6,5,8)(10,16)(11,19)(15,17)(20,24)Polycyclic presentation:
2^8|a^2=h,b^2=h,e^2=fh,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=df,e^b=eh,g^c=gh49 conjugacy classes — view the full character table.
| maximal subgroup of | #250791, #250804, #250823, … |
| maximal normal subgroup of | #250791, #250804, #250823, … |
| maximal quotient of | #6386433, #6388476, #6388642, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55006);