Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23994 |
| automorphism | Aut(G) = C2×#72236881 (order 2048) · Inn(G) = #535 · Out(G) = C25 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #535 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | #2535 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
403 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 403 conjugacy classes of subgroups — too many to draw. Only its 113 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)(3,11)(4,15)(5,18)(6,10)(7,13)(8,16)(9,19)(12,25)(14,23)(17,29)(20,28)(21,27)(22,30)(24,26)(31,32);(1,3)(2,7,10,20)(4,12)(5,13)(6,14)(8,21,22,25)(9,11,18,23)(15,24,29,31)(16,26)(17,27)(19,28)(30,32);(7,20)(11,23)(21,25)(24,31);(1,4,5,16,6,17,19,30)(2,8,9,15,10,22,18,29)(3,12,13,26,14,27,28,32)(7,21,11,24,20,25,23,31)Polycyclic presentation:
2^8|b^2=g,d^2=e,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gh,c^b=cg,f^b=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #500473, #500474, #500743, … |
| maximal normal subgroup of | #500473, #500474, #500743, … |
| maximal quotient of | #316991, #316992, #316993, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21168);