Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2104 |
| automorphism | Aut(G) order 57344 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #401 |
| commutator, Frattini, prime radical, cosocle | C24 | C24 |
| socle | C2 | C2×#401 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11,14,7)(4,8,17,23)(5,9)(6,10)(12,21)(13,25,27,20)(15,19)(16,22,29,32)(18,24)(26,31)(28,30);(1,3)(2,7)(4,15,17,12)(5,13)(6,14)(8,21,23,19)(9,20)(10,11)(16,28,29,26)(18,27)(22,31,32,30)(24,25);(1,4)(2,8,10,23)(3,12)(5,16)(6,17)(7,19,11,21)(9,22,24,32)(13,26)(14,15)(18,29)(20,30,25,31)(27,28);(1,5,6,18)(2,9,10,24)(3,13,14,27)(4,16,17,29)(7,20,11,25)(8,22,23,32)(12,26,15,28)(19,30,21,31)Polycyclic presentation:
2^8|a^2=efg,b^2=ef,c^2=e,d^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=cg,g^b=gh,f^c=fh46 conjugacy classes — view the full character table.
| Fitting subgroup of | #23847⋊C7 |
| maximal subgroup of | #359420, #391183, #401983, … |
| maximal normal subgroup of | #359420, #391183, #401983, … |
| maximal quotient of | #244495, #244497, #244498, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16835);