Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24420 |
| automorphism | Aut(G) = C2×#72236721 (order 4096) · Inn(G) = #2410 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2410 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
650 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 650 conjugacy classes of subgroups — too many to draw. Only its 133 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:
(1,2,7,11,6,4,8,12)(3,9,16,24,15,18,17,13)(5,10,21,25,20,19,22,26)(14,23,29,27,28,31,30,32);(1,3,6,15)(2,9,4,18)(5,14,20,28)(7,17,8,16)(10,23,19,31)(11,13,12,24)(21,30,22,29)(25,32,26,27);(1,4)(2,6)(3,13)(5,19)(7,11)(8,12)(9,17)(10,20)(14,27)(15,24)(16,18)(21,25)(22,26)(23,29)(28,32)(30,31);(1,5)(2,10)(3,14)(4,19)(6,20)(7,22)(8,21)(9,23)(11,26)(12,25)(13,27)(15,28)(16,30)(17,29)(18,31)(24,32)Polycyclic presentation:
2^8|a^2=f,b^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^b=fh,g^b=gh,e^c=eh,f^c=fh,g^c=gh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #501144, #501149, #501150, … |
| maximal normal subgroup of | #501144, #501149, #501150, … |
| maximal quotient of | #267469, #267473, #267478, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17410);