Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | Aut(G) = C2×#72234840 (order 2048) · Inn(G) = #99 · Out(G) = C26 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #99 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×#41 | C23 |
| socle | C2 | C4×#99 |
| prime radical | #532 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
230 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 79 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:
(1,2,6,11,7,4,9,18)(3,12,15,27,16,28,22,32)(5,17,19,31,20,10,24,25)(8,21,23,30,13,26,29,14);(1,3)(2,8)(4,13)(5,14,20,30)(6,15)(7,16)(9,22)(10,12,17,28)(11,23)(18,29)(19,21,24,26)(25,27,31,32);(1,4,6,18,7,2,9,11)(3,13,15,29,16,8,22,23)(5,17,19,31,20,10,24,25)(12,26,27,14,28,21,32,30)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=e,d^2=h,e^2=h,b^a=bd,d^a=dh,f^a=fgh,d^b=dg,g^b=gh,f^d=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #94148, #94149, #94150, … |
| maximal normal subgroup of | #94148, #94149, #94150, … |
| maximal quotient of | #128314, #128340, #128372, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5621);