Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2121 |
| automorphism | Aut(G) = #72310502 (order 2048) · Inn(G) = C2×#99 · Out(G) = #120 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#99 |
| commutator | C23 | C2×C42 |
| Frattini, cosocle | C2×#41 | C23 |
| socle | C2 | C4×#99 |
| prime radical | #525 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
228 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 228 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:
(1,2,6,4,7,10,21,17)(3,11,14,26,15,24,28,29)(5,9,19,16,20,25,32,30)(8,22,12,18,23,31,27,13);(1,3)(2,8)(4,12)(5,18,20,13)(6,14)(7,15)(9,24,25,11)(10,23)(16,29,30,26)(17,27)(19,31,32,22)(21,28);(1,4,6,10,7,17,21,2)(3,12,14,23,15,27,28,8)(5,16,19,25,20,30,32,9)(11,13,26,22,24,18,29,31)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=e,e^2=h,b^a=bd,c^a=ch,f^a=fg,d^b=dg,g^b=gh,f^d=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #150484, #150485, C2×#12624, … |
| maximal normal subgroup of | #150484, #150485, C2×#12624, … |
| maximal quotient of | #128309, #128345, #128367, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5612);