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 |
| prime radical | #3585 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
655 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 655 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:
(1,2,7,21,6,8,5,11)(3,12,20,9,15,23,13,22)(4,16,18,25,19,10,17,26)(14,27,28,31,29,32,30,24);(2,8)(3,13)(5,7)(9,23)(10,16)(12,22)(14,28)(15,20)(17,18)(24,32)(27,31)(29,30);(1,3)(2,9)(4,14)(5,20)(6,15)(7,13)(8,22)(10,24)(11,23)(12,21)(16,31)(17,28)(18,30)(19,29)(25,27)(26,32);(1,4)(2,10)(3,14)(5,18)(6,19)(7,17)(8,16)(9,24)(11,26)(12,27)(13,28)(15,29)(20,30)(21,25)(22,31)(23,32)Polycyclic presentation:
2^8|a^2=fh,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 | #501147, #501148, #501151, … |
| maximal normal subgroup of | #501147, #501148, #501151, … |
| maximal quotient of | #267468, #267472, #267475, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17409);