Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24295 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2408 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
667 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 667 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:
(1,2,8,11)(3,6,14,7)(4,15,19,12)(5,13,22,27)(9,17,24,18)(10,21,25,20)(16,23,31,32)(26,29,28,30);(1,3,6,11,8,14,7,2)(4,9,17,12,19,24,18,15)(5,13,20,25,22,27,21,10)(16,23,29,26,31,32,30,28);(1,4)(2,9)(3,12)(5,16)(6,17)(7,18)(8,19)(10,23)(11,24)(13,26)(14,15)(20,29)(21,30)(22,31)(25,32)(27,28);(1,5,7,21,8,22,6,20)(2,10,14,27,11,25,3,13)(4,16,18,30,19,31,17,29)(9,23,15,28,24,32,12,26)Polycyclic presentation:
2^8|a^2=h,b^2=e,d^2=eh,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,g^b=gh,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #499777, #499779, #499781, … |
| maximal normal subgroup of | #499777, #499779, #499781, … |
| maximal quotient of | #263037, #263039, #263042, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17285);