Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24378 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
404 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 404 conjugacy classes of subgroups — too many to draw. Only its 119 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,15)(12,13)(14,16)(17,18,22,30,21,25,20,24)(19,26,28,23,27,32,29,31);(1,3,4,8)(2,5,6,7)(10,13)(11,16)(19,27)(23,31)(26,32)(28,29);(9,11,12,16)(10,14,13,15)(17,19,21,27)(18,23,25,31)(20,28,22,29)(24,32,30,26);(2,6)(5,7)(10,13)(14,15)(17,20,21,22)(18,24,25,30)(19,28,27,29)(23,32,31,26)Polycyclic presentation:
2^8|a^2=fg,b^2=egh,c^2=g,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#29333, #6461401, #6461417, … |
| maximal normal subgroup of | C2×#29333, #6461401, #6461417, … |
| maximal quotient of | #362801, #364353, #368451, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22321);