Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#121 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| socle | C2 | C22×#121 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,13)(4,18)(5,10)(6,11)(7,12)(8,16)(9,19)(14,28)(15,24)(17,23)(20,25)(21,26)(22,29)(27,30)(31,32);(1,3,5,16,7,15,6,17)(2,8,11,13,12,23,10,24)(4,14,19,30,20,29,21,31)(9,22,18,27,26,28,25,32);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,22)(10,25)(11,18)(12,26)(13,27)(15,29)(16,30)(17,31)(23,28)(24,32);(3,15)(8,23)(13,24)(14,29)(16,17)(22,28)(27,32)(30,31);(1,5,7,6)(2,10,12,11)(3,16,15,17)(4,19,20,21)(8,24,23,13)(9,25,26,18)(14,30,29,31)(22,32,28,27)Polycyclic presentation:
2^8|b^2=f,e^2=h,f^2=h,g^2=h,b^a=bf,c^a=cg,f^a=fh,g^a=gh,d^b=dh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #368852, #368870, #368881, … |
| maximal normal subgroup of | #368852, #368870, #368881, … |
| maximal quotient of | #6353633, #6353636, #6353641, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53457);