Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | Aut(G) = #72303605 (order 4096) · Inn(G) = C2×C22⋊C4 · Out(G) = C2×#392 |
| 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 | |
193 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 193 conjugacy classes of subgroups — too many to draw. Only its 65 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)(9,10,14,18,15,20,27,36,16,21,12,26,13,19,28,35)(11,22,23,39,24,34,40,32,25,37,30,38,29,17,31,33);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,23)(13,29)(14,30)(15,24)(16,25)(18,32)(19,37)(20,22)(21,34)(26,33)(27,31)(28,40)(35,39)(36,38);(9,12,15,28,16,14,13,27)(10,18,20,36,21,26,19,35)(11,23,24,40,25,30,29,31)(17,32,22,38,34,33,37,39)Polycyclic presentation:
2^8|a^2=e,c^2=g,d^2=h,e^2=g,g^2=h,b^a=bd,c^a=ch,d^a=df,d^b=dh,e^b=efh52 conjugacy classes — view the full character table.
| maximal subgroup of | #94398, #128467, #128468, … |
| maximal normal subgroup of | #94398, #128467, #128468, … |
| maximal quotient of | #122402, #122403, #122404, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4457);