Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60367 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1053 conjugacy classes of subgroups; only the 479 normal subgroups are shown.
This group has 1053 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) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,15,20)(11,16,22,30)(12,25,27,34)(13,19,14,18)(17,28,33,26)(21,35,36,40)(23,32,24,31)(29,38,39,37);(2,5)(6,8)(9,11)(10,16)(12,21)(13,23)(14,24)(15,22)(17,29)(18,31)(19,32)(20,30)(25,35)(26,37)(27,36)(28,38)(33,39)(34,40);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,21)(13,26)(14,28)(15,27)(16,29)(18,25)(19,34)(20,33)(22,36)(23,37)(24,38)(30,39)(31,35)(32,40);(11,22)(16,30)(21,36)(23,24)(29,39)(31,32)(35,40)(37,38);(9,13,15,14)(10,18,20,19)(11,23,22,24)(12,26,27,28)(16,31,30,32)(17,25,33,34)(21,37,36,38)(29,35,39,40)Polycyclic presentation:
2^8|a^2=h,c^2=f,e^2=h,g^2=h,b^a=bf,c^a=cg,e^a=eh,g^a=gh,d^b=dh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #147615, #147617, #246332, … |
| maximal normal subgroup of | #147615, #147617, #246332, … |
| maximal quotient of | #6370887, #6371276, #6371386, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53507);