Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | Aut(G) = C25×D42 (order 2048) · Inn(G) = #120 · Out(G) = C26 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #386 |
| prime radical | #3035 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
242 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 242 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) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,6,7,5,3,4,8)(10,14,17,20)(11,18,19,21);(1,3,5,2)(4,8,6,7)(9,10,15,14,13,17,12,20)(11,16,18,22,19,24,21,23);(1,4,5,6)(2,7,3,8)(9,11,13,19)(10,16,17,24)(12,21,15,18)(14,22,20,23)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=g,d^2=g,e^2=h,f^2=gh,b^a=bd,c^a=ce,d^b=dg,e^b=eh,f^b=fg,d^c=dh,e^c=eh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8996, #194366, #194368, … |
| maximal normal subgroup of | C2×#8996, #194366, #194368, … |
| maximal quotient of | #99508, #99510, #99516, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1984);