Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | D42 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| socle | C2 | C2×D42 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1310 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)(4,15)(5,10)(6,11)(7,24)(8,12)(9,18)(14,30)(16,19)(17,31)(20,32)(21,27)(22,28)(23,29)(25,26);(1,3)(2,4)(5,20)(6,15)(7,16)(8,9)(10,26)(11,18)(12,19)(13,24)(14,21)(17,28)(22,25)(23,31)(27,32)(29,30);(1,4)(2,9)(3,12)(5,17)(6,18)(7,13)(8,19)(10,25)(11,16)(14,29)(15,24)(20,27)(21,26)(22,30)(23,32)(28,31);(1,5)(2,10)(3,14)(4,17)(6,21)(7,22)(8,23)(9,25)(11,27)(12,29)(13,30)(15,31)(16,20)(18,26)(19,32)(24,28);(1,6,8,7)(2,11,12,24)(3,15,9,16)(4,18,19,13)(5,21,23,22)(10,27,29,28)(14,31,25,20)(17,26,32,30)Polycyclic presentation:
2^8|e^2=h,f^2=h,g^2=h,b^a=bf,c^a=cg,f^a=fh,g^a=gh,c^b=ch,d^b=df,f^b=fh,g^c=gh,f^d=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #131834, #131835, #131839, … |
| maximal normal subgroup of | #131834, #131835, #131839, … |
| maximal quotient of | #6358256, #6358265, #6358278, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54093);