Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23872 |
| automorphism | Aut(G) = C2×#72236721 (order 4096) · Inn(G) = #2403 · Out(G) = C25 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2403 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
821 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 821 conjugacy classes of subgroups — too many to draw. Only its 141 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:
(1,2)(3,13)(4,9)(5,18)(6,21)(7,12)(8,15)(10,19)(11,23)(14,24)(16,32)(17,25)(20,28)(22,27)(26,31)(29,30);(1,3,7,4)(2,8,12,25)(5,14,20,16)(6,17,23,15)(9,21,13,11)(10,24,27,32)(18,29,28,26)(19,31,22,30);(1,4,7,3)(2,9,12,13)(5,16,20,14)(6,15,23,17)(8,11,25,21)(10,26,27,29)(18,24,28,32)(19,30,22,31);(1,5)(2,10)(3,14)(4,16)(6,22)(7,20)(8,24)(9,26)(11,28)(12,27)(13,29)(15,31)(17,30)(18,21)(19,23)(25,32)Polycyclic presentation:
2^8|b^2=h,c^2=h,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gh,c^b=cg,e^b=eh,e^c=eh,e^d=eh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #152669, #152670, #152703, … |
| maximal normal subgroup of | #152669, #152670, #152703, … |
| maximal quotient of | #244553, #244554, #244557, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16863);