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 |
| prime radical | #3207 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #533 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
848 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)(3,11)(4,8)(5,9)(6,10)(7,13)(12,16)(14,15)(17,18,21,25)(19,26,28,31)(20,24,22,29)(23,30,32,27);(1,3,6,13)(2,7,10,11)(4,12,5,14)(8,15,9,16)(17,18)(19,23)(20,29)(21,25)(22,24)(26,27)(28,32)(30,31);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(17,19)(18,23)(20,27)(21,28)(22,30)(24,31)(25,32)(26,29);(1,5,6,4)(2,9,10,8)(3,12,13,14)(7,15,11,16)(17,20,21,22)(18,24,25,29)(19,27,28,30)(23,31,32,26);(2,10)(7,11)(8,9)(15,16)(18,25)(23,32)(24,29)(26,31)Polycyclic presentation:
2^8|a^2=h,b^2=fh,c^2=fh,d^2=f,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 | #244597, #244598, #244611, … |
| maximal normal subgroup of | #244597, #244598, #244611, … |
| maximal quotient of | #6386326, #6388217, #6388225, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54905);