Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60460 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | #3499 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
999 conjugacy classes of subgroups; only the 479 normal subgroups are shown.
This group has 999 conjugacy classes of subgroups — too many to draw. Only its 479 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) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,16,22)(11,18,25,31)(12,17,24,32)(13,19,27,34)(14,21,15,20)(23,33,37,40)(26,30,38,39)(28,36,29,35);(2,5)(6,8)(9,11,14,12,16,25,15,24)(10,17,20,18,22,32,21,31)(13,23,28,26,27,37,29,38)(19,30,35,33,34,39,36,40);(1,3,4,7)(2,6,5,8)(9,12,14,25,16,24,15,11)(10,18,20,32,22,31,21,17)(13,26,28,37,27,38,29,23)(19,33,35,39,34,40,36,30);(9,13)(10,19)(11,23)(12,26)(14,28)(15,29)(16,27)(17,30)(18,33)(20,35)(21,36)(22,34)(24,38)(25,37)(31,40)(32,39);(13,27)(19,34)(23,37)(26,38)(28,29)(30,39)(33,40)(35,36)Polycyclic presentation:
2^8|a^2=h,b^2=gh,c^2=f,f^2=h,g^2=h,b^a=bf,c^a=cg,f^a=fh,g^a=gh,c^b=ch,e^d=eh58 conjugacy classes — view the full character table.
| Fitting subgroup of | #60627⋊C3 |
| maximal subgroup of | #129975, #152073, #249815, … |
| maximal normal subgroup of | #129975, #152073, #249815, … |
| maximal quotient of | #6371057, #6371700, #6371957, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53615);