Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60768 |
| 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 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
901 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)(19,23)(20,22)(21,25)(24,28)(26,30)(27,29)(31,32);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15);(3,10)(6,9)(11,16)(13,15)(18,22)(20,28)(23,29)(24,25)(27,32)(30,31);(2,7)(6,9)(8,14)(13,15)(17,19)(18,23)(20,27)(21,26)(22,29)(24,31)(25,30)(28,32);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)(19,26)(23,30)(27,32)(29,31)Polycyclic presentation:
2^8|e^2=f,g^2=h,b^a=bf,c^a=cg,d^a=df,g^a=gh,c^b=cf,g^c=gh,e^d=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #131636, #131638, #485011, … |
| maximal normal subgroup of | #131636, #131638, #485011, … |
| maximal quotient of | #6374687, #6375039, #6376028, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53988);