Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D8 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C22 | C2×#102 |
| prime radical | #3169 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
174 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 174 conjugacy classes of subgroups — too many to draw. Only its 61 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,24)(20,23)(21,30)(22,26)(25,28)(27,31)(29,32);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(18,23)(19,21)(20,28)(24,25)(26,32)(27,29)(30,31);(3,9)(6,12)(10,15)(13,16)(17,19,22,21)(18,24,26,30)(20,27,29,28)(23,31,32,25)Polycyclic presentation:
2^8|a^2=e,c^2=h,d^2=fh,e^2=g,f^2=h,b^a=bd,d^a=df,f^a=fh,c^b=cgh,d^b=df,f^b=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #131636, #131642, #131645, … |
| maximal normal subgroup of | #131636, #131642, #131645, … |
| maximal quotient of | #125081, #125126, #125149, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4637);