Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61643 |
| 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 | #3517 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
844 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,9,10,6)(4,12,13,7)(11,14,16,15)(17,18)(19,24)(20,30)(21,27)(22,26)(23,28)(25,31)(29,32);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,19,23,20)(18,24,28,30)(21,31,22,29)(25,26,32,27);(17,20,23,19)(18,25,28,32)(21,31,22,29)(24,26,30,27);(3,10)(6,9)(11,16)(14,15)(17,21,23,22)(18,26,28,27)(19,29,20,31)(24,25,30,32);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(17,21,23,22)(18,27,28,26)(19,29,20,31)(24,32,30,25)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=h,d^2=h,e^2=fh,g^2=h,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gh,d^b=dfh,e^b=eh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #250333, #250336, #250365, … |
| maximal normal subgroup of | #250333, #250336, #250365, … |
| maximal quotient of | #6386473, #6388494, #6388607, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55229);