Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | Aut(G) = #72303605 (order 4096) · Inn(G) = C22⋊C4 · Out(G) = C2×#2010 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | 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×C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
199 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 199 conjugacy classes of subgroups — too many to draw. Only its 69 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)(9,10,14,20,15,21,27,34,16,22,12,18,13,19,28,35)(11,23,24,38,25,33,39,37,26,36,30,40,29,17,31,32);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,24)(13,29)(14,30)(15,25)(16,26)(18,32)(19,36)(20,37)(21,23)(22,33)(27,31)(28,39)(34,40)(35,38);(9,12,15,28,16,14,13,27)(10,18,21,35,22,20,19,34)(11,24,25,39,26,30,29,31)(17,32,23,38,33,37,36,40)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,d^b=dh,e^b=efh76 conjugacy classes — view the full character table.
| maximal subgroup of | #94397, #128446, #128447, … |
| maximal normal subgroup of | #94397, #128446, #128447, … |
| maximal quotient of | #122398, #122399, #122400, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4456);