Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62295 |
| 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 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
857 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)(17,18,22,25)(19,23,21,30)(20,27,29,31)(24,26,32,28);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,19,22,21)(18,23,25,30)(20,28,29,26)(24,31,32,27);(17,20,21,28,22,29,19,26)(18,24,23,27,25,32,30,31);(3,10)(6,9)(11,16)(14,15)(20,29)(24,32)(26,28)(27,31);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(20,29)(24,32)(26,28)(27,31)Polycyclic presentation:
2^8|a^2=fh,b^2=h,c^2=g,e^2=f,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,c^b=ch,d^b=df,d^c=dh,e^c=eh49 conjugacy classes — view the full character table.
| maximal subgroup of | #250440, #250444, #250448, … |
| maximal normal subgroup of | #250440, #250444, #250448, … |
| maximal quotient of | #6386899, #6389523, #6389526, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55500);