Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10208 |
| automorphism | Aut(G) = #72299933 (order 4096) · Inn(G) = #392 · Out(G) = #392 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | #532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
220 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 48 points:
(3,10)(4,5)(7,11)(8,14)(12,13)(15,16)(17,18)(19,29)(20,34)(21,27)(22,26)(23,28)(24,31)(25,35)(30,41)(32,40)(33,39)(36,43)(37,42)(38,46)(44,45)(47,48);(1,2)(3,11)(4,8)(5,14)(6,9)(7,13)(10,15)(12,16)(17,19,23,33)(18,24,28,40)(20,30,36,46)(21,32,22,31)(25,41,42,38)(26,39,27,29)(34,44,43,48)(35,47,37,45);(1,3)(2,7)(4,13)(5,10)(6,12)(8,16)(9,15)(11,14)(17,20,23,36)(18,25,28,42)(19,30,33,46)(21,35,22,37)(24,38,40,41)(26,34,27,43)(29,44,39,48)(31,45,32,47)Polycyclic presentation:
2^8|b^2=g,c^2=g,d^2=g,e^2=gh,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fgh,c^b=cf,d^b=dg,e^b=eg,f^b=fh,d^c=dh,e^c=egh,f^c=fh25 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10210, #232817, #232819, … |
| maximal normal subgroup of | C2×#10210, #232817, #232819, … |
| maximal quotient of | #102688, #102691, #102692, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3198);