Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62972 |
| 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 | |
3399 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,10)(3,11,14,7)(4,15,16,23)(6,9,18,25)(8,22,24,17)(12,27,30,32)(13,20,31,21)(19,29,26,28);(1,3)(2,7)(4,12)(5,14)(6,13)(8,19)(9,21)(10,11)(15,32)(16,30)(17,29)(18,31)(20,25)(22,28)(23,27)(24,26);(1,4)(2,8)(3,12)(5,16)(6,17)(7,19)(9,15)(10,24)(11,26)(13,29)(14,30)(18,22)(20,27)(21,32)(23,25)(28,31);(2,9)(3,13)(4,16)(8,23)(10,25)(12,28)(14,31)(15,24)(17,22)(19,26)(27,32)(29,30);(3,14)(4,17)(7,11)(8,15)(12,28)(13,31)(16,22)(19,27)(20,21)(23,24)(26,32)(29,30);(2,10)(3,13)(7,20)(8,24)(9,25)(11,21)(12,29)(14,31)(15,23)(19,27)(26,32)(28,30)Polycyclic presentation:
2^8|a^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fg,d^b=dh,e^b=eg,f^b=fh,d^c=dg,e^c=eh67 conjugacy classes — view the full character table.
| Fitting subgroup of | #62973⋊C3 |
| maximal subgroup of | C2×#62973, #10586792, #10586817, … |
| maximal normal subgroup of | C2×#62973, #10586792, #10586817, … |
| maximal quotient of | #7627617, #7627618, #7627619, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55961);