Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30496 |
| automorphism | Aut(G) = #72340709 (order 4096) · Inn(G) = #2669 · Out(G) = C25 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2669 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3018 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
490 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 490 conjugacy classes of subgroups — too many to draw. Only its 113 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) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,13,8,14,6,12)(3,15,17,9,18,26,16,27)(4,19,22,10,23,32,21,29)(5,11,25,30,24,31,20,28);(1,3)(2,9)(4,20)(5,21)(6,17)(7,16)(8,18)(10,28)(11,19)(12,26)(13,15)(14,27)(22,24)(23,25)(29,30)(31,32);(1,4)(2,10)(3,5)(6,21)(7,22)(8,23)(9,11)(12,19)(13,32)(14,29)(15,28)(16,20)(17,25)(18,24)(26,30)(27,31);(1,5)(2,11)(3,4)(6,25)(7,20)(8,24)(9,10)(12,30)(13,28)(14,31)(15,32)(16,22)(17,21)(18,23)(19,26)(27,29)Polycyclic presentation:
2^8|a^2=eh,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=de,e^b=eh,g^b=gh,f^c=fh,g^c=gh,e^d=eh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #153148, #153150, #153153, … |
| maximal normal subgroup of | #153148, #153150, #153153, … |
| maximal quotient of | #332840, #332843, #332845, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23485);