Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61591 |
| 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×Q8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
992 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,5)(4,6,8,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24);(2,5)(6,7)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22);(2,6)(4,8)(5,7);(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23);(2,5)(6,7)(10,16)(14,17)(15,24)(22,23)Polycyclic presentation:
2^8|a^2=h,b^2=fh,d^2=fh,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=dfh,g^c=gh52 conjugacy classes — view the full character table.
| Fitting subgroup of | #61843⋊C3 |
| maximal subgroup of | #250841, #250845, #250849, … |
| maximal normal subgroup of | #250841, #250845, #250849, … |
| maximal quotient of | #6386251, #6388007, #6388099, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54831);