Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18132 |
| automorphism | Aut(G) = C22×#72217875 (order 4096) · Inn(G) = C2×#120 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
491 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 491 conjugacy classes of subgroups — too many to draw. Only its 111 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 — 4 generators on 32 points:
(2,8)(4,17)(5,13)(9,25)(10,27)(11,24)(12,19)(14,31)(20,30)(22,23)(26,28)(29,32);(1,2)(3,9)(4,10)(5,19)(6,12)(7,11)(8,13)(14,28)(15,26)(16,23)(17,22)(18,27)(20,32)(21,29)(24,30)(25,31);(1,3,6,15)(2,9,12,26)(4,13,17,5)(7,16,21,18)(8,22,19,10)(11,23,29,27)(14,30,31,20)(24,25,32,28);(1,4,7,14)(2,10,11,28)(3,13,16,30)(5,18,20,15)(6,17,21,31)(8,23,24,9)(12,22,29,25)(19,27,32,26)Polycyclic presentation:
2^8|c^2=g,d^2=h,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,e^b=eg,f^b=fh,e^c=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | #246933, #246935, #246942, … |
| maximal normal subgroup of | #246933, #246935, #246942, … |
| maximal quotient of | #165907, #165909, #165945, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11144);