Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63041 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
3246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,8)(4,7)(5,9)(6,11)(10,15)(12,13)(14,16)(17,18,23,27)(19,20,30,24)(21,25,22,26)(28,31,29,32);(1,2)(3,8)(4,12)(5,9)(6,11)(7,13)(10,14)(15,16)(17,19,23,30)(18,24,27,20)(21,28,22,29)(25,32,26,31);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19)(18,20)(21,28)(22,29)(23,30)(24,27)(25,31)(26,32);(1,4)(2,7)(3,10)(5,13)(6,14)(8,15)(9,12)(11,16)(17,20)(18,19)(21,31)(22,32)(23,24)(25,28)(26,29)(27,30);(1,3)(2,8)(4,10)(5,11)(6,9)(7,15)(12,14)(13,16)(17,21,23,22)(18,25,27,26)(19,28,30,29)(20,31,24,32);(1,2)(3,6)(4,7)(5,9)(8,11)(10,14)(12,13)(15,16)(17,22,23,21)(18,26,27,25)(19,29,30,28)(20,32,24,31)Polycyclic presentation:
2^8|a^2=g,b^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,d^b=dh,f^e=fh70 conjugacy classes — view the full character table.
| Fitting subgroup of | #63047⋊C3 |
| maximal subgroup of | #347436, #347456, #347476, … |
| maximal normal subgroup of | #347436, #347456, #347476, … |
| maximal quotient of | #7627405, #7627406, #7627426, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 56035);