Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23882 |
| automorphism | Aut(G) = C2×#72236721 (order 4096) · Inn(G) = #2403 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2403 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
848 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 848 conjugacy classes of subgroups — too many to draw. Only its 141 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)(4,17)(5,10)(6,11)(7,12)(8,15)(9,19)(14,22)(16,27)(18,26)(20,25)(21,23)(24,30)(28,29)(31,32);(1,3)(2,8)(4,14)(5,21)(6,15)(7,16)(9,24)(10,13)(11,27)(12,23)(17,28)(18,31)(19,29)(20,30)(22,26)(25,32);(1,4,7,20)(2,9,12,26)(3,14,16,30)(5,19,6,18)(8,22,23,24)(10,25,11,17)(13,28,27,32)(15,31,21,29);(2,10)(5,6)(8,13)(9,25)(11,12)(15,21)(17,26)(18,19)(22,28)(23,27)(24,32)(29,31)Polycyclic presentation:
2^8|c^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,g^a=gh,c^b=cg,e^b=eh,e^c=eh,f^c=fh,e^d=eh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #501193, #501195, #501201, … |
| maximal normal subgroup of | #501193, #501195, #501201, … |
| maximal quotient of | #244587, #244588, #244589, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16871);