Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61106 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×D42 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,14)(4,10)(5,13)(6,12)(7,11)(8,17)(9,19)(15,23)(16,30)(18,29)(20,28)(21,25)(22,26)(24,32)(27,31);(1,3)(2,8)(4,15)(5,21)(6,17)(7,18)(9,23)(10,27)(11,25)(12,29)(13,14)(16,19)(20,32)(22,31)(24,26)(28,30);(1,4)(2,9)(3,15)(5,19)(6,22)(7,20)(8,23)(10,12)(11,26)(13,28)(14,30)(16,21)(17,31)(18,32)(24,25)(27,29);(1,4)(2,10)(3,16)(5,22)(6,19)(7,20)(8,24)(9,12)(11,28)(13,26)(14,30)(15,21)(17,32)(18,31)(23,25)(27,29);(2,11)(8,25)(9,26)(10,28)(12,13)(14,29)(23,24)(27,30)Polycyclic presentation:
2^8|f^2=h,g^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gh,d^b=df,f^b=fh,g^c=gh,f^d=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #499777, #499780, #499781, … |
| maximal normal subgroup of | #499777, #499780, #499781, … |
| maximal quotient of | #6358426, #6358431, #6358433, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54094);