Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #447 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#472 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
256 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 256 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,8,14,9,15,24,29,11,17,25,28,10,16,22,12)(7,18,19,35,26,13,30,31,21,34,23,27,20,32,36,33);(1,2,3,4)(6,12)(8,22)(9,10)(13,27)(14,16)(15,28)(17,29)(18,33)(19,36)(20,26)(23,30)(24,25)(31,34)(32,35);(5,7)(6,13)(8,23)(9,26)(10,20)(11,21)(12,27)(14,33)(15,34)(16,18)(17,32)(19,25)(22,30)(24,36)(28,31)(29,35)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=gh,e^2=h,f^2=gh,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×#12996, #494454, #494455, … |
| maximal normal subgroup of | C2×#12996, #494454, #494455, … |
| maximal quotient of | #136607, #136635, #136878, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5984);