Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60756 |
| 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 | #3494 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
850 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)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(18,23)(19,27)(20,22)(24,31)(28,32)(29,30);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,18,21,23)(19,28,29,24)(20,25,22,26)(27,31,30,32);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,19,21,29)(18,24,23,28)(20,27,22,30)(25,32,26,31);(2,8)(6,9)(7,16)(14,15)(17,20,21,22)(18,25,23,26)(19,27,29,30)(24,32,28,31);(18,23)(24,28)(25,26)(31,32)Polycyclic presentation:
2^8|b^2=f,c^2=f,d^2=h,g^2=h,b^a=bf,c^a=cg,d^a=df,g^a=gh,c^b=cf,e^b=eh,g^c=gh49 conjugacy classes — view the full character table.
| maximal subgroup of | #6669696, #6669697, #6669708, … |
| maximal normal subgroup of | #6669696, #6669697, #6669708, … |
| maximal quotient of | #6374371, #6374546, #6375011, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53863);