Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | Aut(G) = #72312248 (order 2048) · Inn(G) = #447 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #447 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1874 |
| prime radical | #3067 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
238 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 73 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,7,14,8,15,32,46,9,16,31,45,6,13,5,12)(3,17,21,39,22,43,19,36,23,50,29,10,20,37,57,38)(4,24,27,44,26,42,30,48,28,47,34,40,33,11,25,41)(18,51,54,62,53,61,52,63,55,64,59,60,58,49,56,35);(1,3,8,20,9,23,6,22)(2,10,15,36,16,39,13,38)(4,18,26,58,28,55,33,53)(5,29,7,19,32,21,31,57)(11,35,24,60,42,63,47,62)(12,43,14,17,46,37,45,50)(25,59,27,52,30,54,34,56)(40,64,41,61,44,51,48,49);(1,4)(2,11)(3,18)(5,30)(6,33)(7,34)(8,26)(9,28)(10,35)(12,44)(13,47)(14,48)(15,24)(16,42)(17,49)(19,56)(20,58)(21,59)(22,53)(23,55)(25,32)(27,31)(29,54)(36,60)(37,64)(38,62)(39,63)(40,46)(41,45)(43,51)(50,61)(52,57)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=gh,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,d^b=dg,e^b=eh,f^b=fg,g^b=gh,d^c=dh,e^c=eh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12995, #494451, #495719, … |
| maximal normal subgroup of | C2×#12995, #494451, #495719, … |
| maximal quotient of | #136607, #136608, #136635, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5983);