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 | |
831 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 831 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)(3,14)(4,19)(5,22)(6,12)(7,11)(8,13)(9,17)(10,21)(15,24)(16,25)(18,28)(20,27)(23,26)(29,31)(30,32);(1,3)(2,9)(4,15)(5,16)(6,23)(7,17)(8,18)(10,25)(11,28)(12,14)(13,26)(19,29)(20,32)(21,31)(22,30)(24,27);(1,4,8,5)(2,10,13,27)(3,15,18,16)(6,21,7,20)(9,24,26,25)(11,19,12,22)(14,29,28,30)(17,32,23,31);(1,5,8,4)(2,10,13,27)(3,16,18,15)(6,20,7,21)(9,25,26,24)(11,19,12,22)(14,30,28,29)(17,31,23,32)Polycyclic presentation:
2^8|c^2=h,d^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 | #501192, #501194, #501200, … |
| maximal normal subgroup of | #501192, #501194, #501200, … |
| maximal quotient of | #244591, #244592, #244593, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16873);