Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13096 |
| automorphism | Aut(G) = C2×#72236702 (order 2048) · Inn(G) = #1898 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1898 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×#41 | C23 |
| prime radical | C2×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
336 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 336 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,9,7,12,5,11)(3,13,17,29,18,31,16,30)(4,19,14,27,21,10,22,24)(8,23,25,20,26,32,28,15);(1,3,14,23)(2,8,24,29)(4,20,6,16)(5,17,21,15)(7,18,22,32)(9,25,10,13)(11,28,19,31)(12,26,27,30);(2,9)(4,21)(5,6)(8,25)(10,24)(11,12)(13,29)(15,20)(16,17)(19,27)(26,28)(30,31)Polycyclic presentation:
2^8|a^2=f,b^2=df,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dg,e^b=eh,f^b=fg,g^b=gh,d^c=dh,e^c=eh,f^c=fh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13100, #494757, #494758, … |
| maximal normal subgroup of | C2×#13100, #494757, #494758, … |
| maximal quotient of | #137684, #137690, #137732, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6088);