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 | #3548 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
862 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,22,21)(19,26,25,23)(20,27,29,32)(24,31,28,30);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,30)(29,32);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20)(18,24)(19,27)(21,28)(22,29)(23,31)(25,32)(26,30);(3,11)(6,9)(10,16)(13,15)(17,21,22,18)(19,26,25,23)(20,28,29,24)(27,30,32,31);(2,8)(6,9)(7,14)(13,15)(18,21)(19,25)(24,28)(27,32)Polycyclic presentation:
2^8|a^2=h,c^2=fh,d^2=h,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=dfh,e^b=eh,g^c=gh,e^d=eh49 conjugacy classes — view the full character table.
| maximal subgroup of | #244589, #244590, #244603, … |
| maximal normal subgroup of | #244589, #244590, #244603, … |
| maximal quotient of | #6386193, #6387947, #6388217, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54798);