Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11855 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#102 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
167 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 167 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) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9,11,6)(4,7)(5,8)(10,15,12,14)(13,16)(18,24,26,29)(19,20)(21,22)(25,27,32,28)(30,31);(1,3)(2,6)(4,12)(5,11)(7,15)(8,9)(10,13)(14,16)(17,18,23,26)(19,29,31,24)(20,32,30,25)(21,28,22,27);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,19,22,20,23,31,21,30)(18,25,28,24,26,32,27,29)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=fh,d^2=gh,f^2=h,g^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gh,c^b=cf,d^b=dg,e^b=efgh,f^b=fh,g^b=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #131715, #131716, #131717, … |
| maximal normal subgroup of | #131715, #131716, #131717, … |
| maximal quotient of | #124298, #124452, #124472, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4845);