Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31282 |
| automorphism | Aut(G) = #72338516 (order 4096) · Inn(G) = #2684 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2684 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
377 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 377 conjugacy classes of subgroups — too many to draw. Only its 97 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:
(1,2,7,12)(3,13,17,24)(4,18,21,28)(5,11,6,10)(8,15,25,16)(9,22,27,20)(14,26,19,23)(29,30,32,31);(1,3,6,15,7,17,5,16)(2,8,11,24,12,25,10,13)(4,19,20,30,21,14,22,31)(9,26,18,32,27,23,28,29);(1,4)(2,9)(3,14)(5,20)(6,22)(7,21)(8,23)(10,18)(11,28)(12,27)(13,29)(15,30)(16,31)(17,19)(24,32)(25,26);(3,15)(5,6)(8,24)(10,11)(13,25)(14,30)(16,17)(18,28)(19,31)(20,22)(23,32)(26,29)Polycyclic presentation:
2^8|a^2=h,b^2=f,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fh,c^b=cg,d^b=de,g^b=gh,f^c=fh,e^d=eh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #153201, #153202, #153205, … |
| maximal normal subgroup of | #153201, #153202, #153205, … |
| maximal quotient of | #345820, #345822, #345824, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24272);