Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×C22⋊C8 |
| prime radical | C22×#41 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
264 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 264 conjugacy classes of subgroups — too many to draw. Only its 73 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 — 3 generators on 40 points:
(1,2)(3,7,8,5)(4,6)(9,10,13,18,14,19,28,36,15,20,29,37,12,17,26,34)(11,21,24,38,23,31,39,35,22,33,27,40,25,16,30,32);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,25)(13,27)(14,23)(15,22)(17,33)(18,35)(19,21)(20,31)(24,29)(26,39)(28,30)(32,37)(34,38)(36,40);(11,22)(16,31)(21,33)(23,25)(24,27)(30,39)(32,35)(38,40)Polycyclic presentation:
2^8|a^2=e,d^2=h,e^2=g,g^2=h,b^a=bd,d^a=df,c^b=ch,d^b=dh,e^b=efh64 conjugacy classes — view the full character table.
| maximal subgroup of | #129205, #129230, #129285, … |
| maximal normal subgroup of | #129205, #129230, #129285, … |
| maximal quotient of | #122270, #122271, #122272, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4410);