Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62243 |
| 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 | |
931 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18)(19,23)(20,27)(21,30)(22,25)(24,28)(26,31)(29,32);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(17,19,22,21)(18,23,25,30)(20,26,29,28)(24,27,31,32);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20,21,28,22,29,19,26)(18,24,23,27,25,31,30,32);(3,12)(7,10)(11,13)(15,16);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)Polycyclic presentation:
2^8|b^2=h,c^2=fg,e^2=f,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=df64 conjugacy classes — view the full character table.
| maximal subgroup of | #250095, #250097, #250099, … |
| maximal normal subgroup of | #250095, #250097, #250099, … |
| maximal quotient of | #6386938, #6389622, #6389626, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55515);