Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7225 |
| automorphism | Aut(G) = #72281426 (order 2048) · Inn(G) = H(3,ℤ/4) · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | H(3,ℤ/4) |
| commutator | C2×C8 | C42 |
| Frattini, cosocle | C2×#131 | C22 |
| socle | C2 | #1289 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
124 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2)(3,13)(4,18,20,44)(5,23)(6,11)(7,12)(8,14,26,38)(9,22,29,46)(10,31)(15,41)(16,35)(17,36)(19,32)(21,48,47,30)(24,33)(25,51)(27,54,52,62)(28,40,53,59)(34,55)(37,57)(39,61,60,50)(42,58)(43,63)(45,49,64,56);(1,3,5,15,6,16,24,42,7,17,25,43,19,37,31,55)(2,8,10,27,32,53,51,61,12,26,33,52,11,28,23,50)(4,14,21,39,22,40,49,62,20,38,47,60,46,59,56,54)(9,13,30,34,44,57,64,63,29,36,48,58,18,35,45,41)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fgh,e^2=fg,f^2=h,b^a=bc,c^a=cf,e^a=efgh,g^a=gh,d^b=dg,d^c=dh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7227, #127342, #127355, … |
| maximal normal subgroup of | C2×#7227, #127342, #127355, … |
| maximal quotient of | #93011, #93012, #93013, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 215);