Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | Aut(G) = #72338421 (order 8192) · Inn(G) = D42 · Out(G) = #1595 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | D42 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | #2366 |
| prime radical | #3150 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
440 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 440 conjugacy classes of subgroups — too many to draw. Only its 165 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)(3,13,18,29)(4,19)(5,9)(6,10)(7,12)(8,16,26,17)(11,23)(14,31,15,24)(20,27)(21,30,32,25)(22,28);(1,3,6,17,7,18,23,16)(2,8,10,29,12,26,11,13)(4,14,5,21,20,15,22,32)(9,24,27,25,28,31,19,30);(1,4,6,5,7,20,23,22)(2,9,11,19,12,28,10,27)(3,14,17,21,18,15,16,32)(8,24,13,30,26,31,29,25);(1,5,6,20,7,22,23,4)(2,9,10,27,12,28,11,19)(3,15,16,21,18,14,17,32)(8,25,13,24,26,30,29,31)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=f,d^2=e,e^2=h,f^2=h,b^a=be,c^a=cf,f^a=fh,d^b=de,e^b=eh,g^b=gh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #152476, #152579, #152590, … |
| maximal normal subgroup of | #152476, #152579, #152590, … |
| maximal quotient of | #231470, #231471, #231478, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16185);