Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23924 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
335 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 335 conjugacy classes of subgroups — too many to draw. Only its 115 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)(3,8)(4,6)(5,7)(9,10,12,14)(11,13,15,16)(17,18,23,27)(19,24,28,25)(20,29,30,32)(21,26,22,31);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,22)(21,30)(23,28)(25,27)(26,32)(29,31);(1,4,5,3)(2,6,7,8)(11,15)(13,16)(17,20,23,30)(18,25,27,24)(19,21,28,22)(26,32,31,29);(2,7)(6,8)(17,21,23,22)(18,26,27,31)(19,20,28,30)(24,29,25,32)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=eh,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#27621, #6458906, #6458911, … |
| maximal normal subgroup of | C2×#27621, #6458906, #6458911, … |
| maximal quotient of | #359932, #364048, #366406, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20609);