Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1408 |
| automorphism | Aut(G) = #72319331 (order 4096) · Inn(G) = #353 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #353 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C4⋊Q8 | C23 |
| socle | C2 | C2×#353 |
| prime radical | #3147 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
204 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 204 conjugacy classes of subgroups — too many to draw. Only its 45 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,10,6,11,21,28)(3,12,14,27,16,23,30,31)(4,8,17,25,18,9,20,26)(7,19,22,13,24,29,32,15);(1,3)(2,7)(4,13)(5,19)(6,16)(8,12)(9,23)(10,27)(11,24)(14,17)(15,18)(20,30)(21,29)(22,26)(25,32)(28,31);(1,4,6,18)(2,8,11,9)(3,13,16,15)(5,17,21,20)(7,12,24,23)(10,25,28,26)(14,29,30,19)(22,27,32,31)Polycyclic presentation:
2^8|a^2=e,c^2=h,d^2=gh,e^2=h,f^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,d^b=dgh,e^b=efg,f^b=fh,e^d=egh,f^e=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #94778, #94782, #94797, … |
| maximal normal subgroup of | #94778, #94782, #94797, … |
| maximal quotient of | #149573, #149623, #149689, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6577);