Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24986 |
| automorphism | Aut(G) = #72337551 (order 2048) · Inn(G) = #555 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #555 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | #2495 |
| prime radical | C2×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
219 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 219 conjugacy classes of subgroups — too many to draw. Only its 97 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:
(1,2,8,12,7,14,6,13)(3,15,24,26,17,11,5,9)(4,18,22,10,23,29,21,30)(16,25,19,27,31,32,20,28);(1,3,6,5,7,17,8,24)(2,9,12,15,14,26,13,11)(4,19,22,31,23,20,21,16)(10,27,18,25,30,28,29,32);(1,4)(2,10)(3,16)(5,20)(6,21)(7,23)(8,22)(9,25)(11,28)(12,29)(13,18)(14,30)(15,27)(17,31)(19,24)(26,32);(1,5,6,17,7,24,8,3)(2,11,12,9,14,15,13,26)(4,20,21,31,23,19,22,16)(10,28,29,25,30,27,18,32)Polycyclic presentation:
2^8|a^2=efg,b^2=f,d^2=e,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,d^b=dh,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #505034, #505035, #505520, … |
| maximal normal subgroup of | #505034, #505035, #505520, … |
| maximal quotient of | #360422, #362693, #365249, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20822);