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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
343 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 343 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)(3,7)(4,15)(5,10)(6,18)(8,17)(9,11)(12,28)(13,22)(14,24)(16,26)(19,23)(20,27)(21,29)(25,31)(30,32);(1,3,11,13,5,14,6,19)(2,7,18,23,10,24,9,22)(4,12,20,29,16,30,17,31)(8,25,26,32,27,21,15,28);(1,4,5,16)(2,8,10,27)(3,12)(6,20,11,17)(7,21)(9,26,18,15)(13,31)(14,30)(19,29)(22,28)(23,32)(24,25);(2,9)(6,11)(7,22)(8,26)(10,18)(13,19)(15,27)(17,20)(21,28)(23,24)(25,32)(29,31)Polycyclic presentation:
2^8|b^2=ef,c^2=e,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 | #6792405, #6792407, #6792444, … |
| maximal normal subgroup of | #6792405, #6792407, #6792444, … |
| maximal quotient of | #345968, #345971, #345980, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24406);