Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3186 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| socle | C22 | C23×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
923 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18)(19,27)(20,26)(21,23)(22,25)(24,29)(28,32)(30,31);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(18,23)(19,28)(25,26)(29,30);(3,10)(6,9)(11,16)(13,15)(17,19)(18,24)(20,29)(21,28)(22,30)(23,31)(25,27)(26,32);(2,7)(6,9)(8,14)(13,15)(17,20,21,22)(18,25,23,26)(19,29,28,30)(24,27,31,32);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)Polycyclic presentation:
2^8|d^2=h,e^2=fh,g^2=h,b^a=bf,c^a=cg,d^a=df,g^a=gh,c^b=cf,g^c=gh64 conjugacy classes — view the full character table.
| maximal subgroup of | #111278, #111286, #111313, … |
| maximal normal subgroup of | #111278, #111286, #111313, … |
| maximal quotient of | #6374643, #6375030, #6375959, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53936);