Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12721 |
| automorphism | Aut(G) = C2×#72238017 (order 4096) · Inn(G) = #1782 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1782 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×#41 | C23 |
| prime radical | #535 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
347 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 347 conjugacy classes of subgroups — too many to draw. Only its 69 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)(3,12,15,27)(4,9,17,25)(5,10)(6,11)(7,13)(8,16,24,31)(14,20,18,32)(19,23)(21,28)(22,29)(26,30);(1,3)(2,7)(4,13)(5,18)(6,15)(8,20)(9,12)(10,26)(11,22)(14,19)(16,30)(17,29)(21,23)(24,32)(25,27)(28,31);(1,4,6,17)(2,8)(3,13,15,29)(5,16,19,31)(7,20)(9,10)(11,24)(12,26)(14,28,18,30)(21,27)(22,32)(23,25)Polycyclic presentation:
2^8|a^2=f,c^2=g,e^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,e^b=eh,f^b=fg,d^c=dh,f^c=fh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #94451, #152145, #152146, … |
| maximal normal subgroup of | #94451, #152145, #152146, … |
| maximal quotient of | #129291, #129299, #129371, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5709);