Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2180 |
| automorphism | Aut(G) = #72282918 (order 2048) · Inn(G) = #442 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #442 |
| commutator | C8 | C2×C42 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1776 |
| prime radical | C23 | #137 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
146 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 85 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:
(1,2,5,12,7,13,21,11,8,14,22,28,6,10,4,20)(3,15,17,26,18,27,31,30,19,29,32,9,23,25,16,24);(1,3,6,18,8,19,7,23)(2,9,10,24,14,26,13,30)(4,17,22,31,21,32,5,16)(11,27,12,29,20,25,28,15);(2,10,14,13)(4,21)(5,22)(9,24,26,30)(11,28,20,12)(15,25,29,27)(16,31)(17,32)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=de,d^2=gh,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=cg,d^a=dh,d^b=dg,e^b=egh,g^b=gh,d^c=dh,e^c=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12678, #493162, #493247, … |
| maximal normal subgroup of | C2×#12678, #493162, #493247, … |
| maximal quotient of | #129120, #129124, #129132, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5666);