Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63011 |
| 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 | |
3308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,7,5)(3,10,8,4)(6,9,14,13)(11,16,15,12)(17,18,22,25)(19,26,27,23)(20,21,24,30)(28,32,31,29);(1,3)(2,4)(5,10)(6,11)(7,8)(9,12)(13,16)(14,15)(19,27)(23,26)(28,31)(29,32);(1,3)(2,4)(5,10)(6,11)(7,8)(9,12)(13,16)(14,15)(17,19)(18,23)(20,28)(21,29)(22,27)(24,31)(25,26)(30,32);(1,4,7,10)(2,8,5,3)(6,12,14,16)(9,15,13,11);(1,5,7,2)(3,10,8,4)(6,9,14,13)(11,12,15,16)(17,20)(18,21)(19,28)(22,24)(23,29)(25,30)(26,32)(27,31);(1,6,7,14)(2,9,5,13)(3,11,8,15)(4,12,10,16)(17,21)(18,24)(19,29)(20,25)(22,30)(23,31)(26,28)(27,32)Polycyclic presentation:
2^8|a^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,f^e=fh76 conjugacy classes — view the full character table.
| maximal subgroup of | #346948, #347195, #347209, … |
| maximal normal subgroup of | #346948, #347195, #347209, … |
| maximal quotient of | #7627222, #7627234, #7627243, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 56019);