Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23924 |
| automorphism | Aut(G) order 65536 |
| 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 | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
444 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 444 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,13,15,16)(17,18,22,27)(19,29,20,24)(21,28,23,26)(25,30,32,31);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,20)(18,24,27,29)(21,31,23,30)(25,28,32,26);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,20,22,19)(18,25,27,32)(21,31,23,30)(24,28,29,26);(2,5)(6,8)(17,21,22,23)(18,26,27,28)(19,30,20,31)(24,32,29,25)Polycyclic presentation:
2^8|a^2=gh,b^2=h,c^2=e,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 | #144425, #144427, #144431, … |
| maximal normal subgroup of | #144425, #144427, #144431, … |
| maximal quotient of | #359822, #366406, #370007, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20597);