Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C8 | C24 |
| commutator, socle | C22 | C23×C8 |
| prime radical | #532 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
278 conjugacy classes of subgroups; only the 177 normal subgroups are shown.
This group has 278 conjugacy classes of subgroups — too many to draw. Only its 177 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 — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,12,17,13,18,23,31,14,19,24,32,25,33,36,40)(11,15,21,26,22,27,35,37,20,16,28,29,34,30,38,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,21)(13,22)(14,20)(16,19)(17,26)(18,27)(23,35)(24,28)(25,34)(29,32)(30,33)(31,37)(36,38)(39,40);(3,8)(5,7)(9,11)(10,16)(12,21)(13,22)(14,20)(15,19)(17,29)(18,30)(23,35)(24,28)(25,34)(26,32)(27,33)(31,39)(36,38)(37,40);(2,6)(5,7)(11,20)(15,16)(21,28)(22,34)(26,29)(27,30)(35,38)(37,39)Polycyclic presentation:
2^8|a^2=f,f^2=g,g^2=h,b^a=be,c^a=ch,d^a=de,c^b=ce,d^b=dh,d^c=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | #135858, #135947, #135948, … |
| maximal normal subgroup of | #135858, #135947, #135948, … |
| maximal quotient of | #493270, #493280, #493307, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26482);