Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32696 |
| automorphism | Aut(G) = C2×#72236189 (order 2048) · Inn(G) = #2749 · Out(G) = C24 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2749 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
276 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 276 conjugacy classes of subgroups — too many to draw. Only its 89 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:
(1,2,5,10,6,9,20,26)(3,11,14,27,13,7,19,22)(4,15,18,31,17,8,23,25)(12,28,16,30,29,21,32,24);(1,3)(2,7)(4,16)(5,19)(6,13)(8,24)(9,11)(10,27)(12,18)(14,20)(15,30)(17,32)(21,25)(22,26)(23,29)(28,31);(1,4)(2,8,9,15)(3,12,13,29)(5,18)(6,17)(7,21)(10,25,26,31)(11,28)(14,16,19,32)(20,23)(22,24)(27,30);(2,9)(5,20)(8,15)(14,19)(16,32)(18,23)(22,27)(24,30)Polycyclic presentation:
2^8|a^2=g,c^2=ef,g^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,c^b=cg,d^b=de,f^b=fh,g^b=gh,e^c=eh,f^c=fh,g^d=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | #6803849, #6803850, #6803853, … |
| maximal normal subgroup of | #6803849, #6803850, #6803853, … |
| maximal quotient of | #384264, #384295, #384427, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25684);