Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31418 |
| automorphism | Aut(G) = C2×#72236189 (order 2048) · Inn(G) = #2688 · Out(G) = C24 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2688 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3068 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
335 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 335 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)(3,12)(4,16)(5,9)(6,11)(7,14)(8,17)(10,21)(13,28)(15,24)(18,27)(19,23)(20,26)(22,31)(25,30)(29,32);(1,3,5,19,6,14,21,15)(2,7,10,23,11,12,9,24)(4,13,17,31,18,29,20,30)(8,25,16,32,26,22,27,28);(1,4,6,18)(2,8,11,26)(3,13)(5,20,21,17)(7,22)(9,16,10,27)(12,25)(14,29)(15,31)(19,30)(23,32)(24,28);(2,9)(3,14)(5,21)(7,23)(8,16)(10,11)(12,24)(13,29)(17,20)(22,32)(25,28)(26,27)Polycyclic presentation:
2^8|b^2=ef,c^2=eh,f^2=h,b^a=be,c^a=cf,d^a=def,f^a=fh,g^a=gh,c^b=cg,d^b=de,e^b=eh,f^b=fh,g^b=gh,f^c=fh,g^c=gh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #6792404, #6792408, #6792445, … |
| maximal normal subgroup of | #6792404, #6792408, #6792445, … |
| maximal quotient of | #345972, #345975, #345976, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24408);