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 | |
3385 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)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,18,22,25)(19,21,27,23)(20,24,28,31)(26,29,32,30);(3,10)(4,12)(7,9)(8,13)(17,19,22,27)(18,23,25,21)(20,26,28,32)(24,30,31,29);(2,6)(7,9)(8,13)(14,15)(20,28)(24,31)(26,32)(29,30);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,20,22,28)(18,24,25,31)(19,26,27,32)(21,29,23,30);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,19)(18,21)(20,26)(22,27)(23,25)(24,29)(28,32)(30,31);(1,4)(2,8)(3,11)(5,12)(6,13)(7,14)(9,15)(10,16)(17,21)(18,19)(20,29)(22,23)(24,26)(25,27)(28,30)(31,32)Polycyclic presentation:
2^8|a^2=g,b^2=g,d^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 | #62976⋊C3 |
| maximal subgroup of | #347038, #347419, C2×#62976, … |
| maximal normal subgroup of | #347038, #347419, C2×#62976, … |
| maximal quotient of | #7627619, #7627623, #7627625, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55964);