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) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22⋊C4 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×C22⋊C8 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
157 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 157 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) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,21,26,22,27,36,44,23,28,37,45,20,25,34,42)(19,29,31,46,30,40,47,43,32,41,35,48,33,24,38,39);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,24)(20,33)(21,35)(22,30)(23,32)(25,41)(26,43)(27,29)(28,40)(31,37)(34,47)(36,38)(39,45)(42,46)(44,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)Polycyclic presentation:
2^8|a^2=e,c^2=f,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 | #128434, #128435, #128436, … |
| maximal normal subgroup of | #128434, #128435, #128436, … |
| maximal quotient of | #124099, #124130, #124291, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4450);