Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
656 conjugacy classes of subgroups; only the 405 normal subgroups are shown.
This group has 656 conjugacy classes of subgroups — too many to draw. Only its 405 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,6)(7,8)(12,20)(15,22)(18,24)(21,23)(25,26,29,32)(27,33,35,30)(28,36,37,31)(34,38,40,39);(9,10,13,16)(11,14,19,17)(12,15,20,22)(18,21,24,23)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,37)(39,40);(1,2)(3,7)(4,8)(5,6)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,27,29,35)(26,30,32,33)(28,34,37,40)(31,38,36,39);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,37)(38,40);(1,4,5,3)(2,8,6,7)(9,12)(10,15)(11,18)(13,20)(14,21)(16,22)(17,23)(19,24)(25,28,29,37)(26,31,32,36)(27,34,35,40)(30,38,33,39)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=f,d^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^c=dfh52 conjugacy classes — view the full character table.
| Fitting subgroup of | #37603⋊C3 |
| maximal subgroup of | #97015, #241616, #241620, … |
| maximal normal subgroup of | #97015, #241616, #241620, … |
| maximal quotient of | #556533, #556541, #556735, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30591);