Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | Aut(G) order 98304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
861 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,5)(4,7)(6,8)(10,13)(14,16)(18,22)(19,25)(24,29)(26,31);(2,3)(6,7)(11,15)(14,16)(18,22)(23,28)(24,29)(30,32);(1,2,5,3)(4,6,8,7)(9,10,12,13)(11,14,15,16)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32);(1,3,5,2)(4,7,8,6)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32);(1,4)(2,6)(3,7)(5,8)(9,11,12,15)(10,14,13,16)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)Polycyclic presentation:
2^8|c^2=fgh,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=df,e^b=egh46 conjugacy classes — view the full character table.
| Fitting subgroup of | #53380⋊C3 |
| maximal subgroup of | C2×#53380, #8105852, #8105895, … |
| maximal normal subgroup of | C2×#53380, #8105852, #8105895, … |
| maximal quotient of | #606820, #607029, #607030, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46368);