Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | Aut(G) = #120×#525 (order 2048) · Inn(G) = C2×C22⋊C4 · Out(G) = C22×#41 |
| 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 | #525 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
196 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 67 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)(9,10,14,20,15,21,28,35,16,22,12,18,13,19,29,36)(11,23,26,39,25,34,40,32,27,37,24,38,30,17,31,33);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,26)(13,30)(14,24)(15,25)(16,27)(18,33)(19,37)(20,32)(21,23)(22,34)(28,31)(29,40)(35,38)(36,39);(9,12,15,29,16,14,13,28)(10,18,21,36,22,20,19,35)(11,24,25,31,27,26,30,40)(17,32,23,38,34,33,37,39)Polycyclic presentation:
2^8|a^2=e,c^2=g,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 | #129569, #129570, #129597, … |
| maximal normal subgroup of | #129569, #129570, #129597, … |
| maximal quotient of | #122406, #122407, #122408, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4458);