Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11858 |
| automorphism | Aut(G) = #72306287 (order 4096) · Inn(G) = C2×#102 · Out(G) = C22×#41 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#102 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C4×C8 | C23 |
| prime radical | #466 | C4 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
153 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 153 conjugacy classes of subgroups — too many to draw. Only its 57 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,6)(3,4)(7,8)(9,10,13,19,14,20,28,34,15,17,29,35,27,36,12,18)(11,21,23,31,24,37,22,38,25,39,26,16,30,32,40,33);(1,2)(3,6)(4,8)(5,7)(9,11)(10,16)(12,26)(13,22)(14,30)(15,25)(17,31)(18,37)(19,21)(20,38)(23,28)(24,27)(29,40)(32,34)(33,36)(35,39);(1,3,5,4)(2,6,7,8)(10,17)(16,31)(18,34)(19,35)(20,36)(21,39)(32,37)(33,38)Polycyclic presentation:
2^8|a^2=e,c^2=f,d^2=gh,e^2=fg,g^2=h,b^a=bd,c^a=cfh,d^a=df,d^b=dg,e^b=efg,g^b=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #130434, #130444, #130445, … |
| maximal normal subgroup of | #130434, #130444, #130445, … |
| maximal quotient of | #124300, #124302, #124427, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4967);