Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31077 |
| automorphism | Aut(G) = #72252039 (order 2048) · Inn(G) = H(3,𝔽4) · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | H(3,𝔽4) |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | #2684 |
| prime radical | #3018 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
370 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 370 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)(3,13)(4,19)(5,11)(6,10)(7,12)(8,16)(9,21)(14,29)(15,18)(17,25)(20,30)(22,27)(23,28)(24,32)(26,31);(1,3,6,17,7,18,5,16)(2,8,11,15,12,25,10,13)(4,20,21,14,23,31,22,32)(9,26,19,29,27,30,28,24);(1,4)(2,9)(3,14)(5,21)(6,22)(7,23)(8,24)(10,19)(11,28)(12,27)(13,26)(15,30)(16,31)(17,20)(18,32)(25,29);(1,2)(3,15)(4,9)(5,11)(6,10)(7,12)(8,17)(13,18)(14,30)(16,25)(19,22)(20,24)(21,28)(23,27)(26,32)(29,31)Polycyclic presentation:
2^8|b^2=f,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fh,c^b=cg,d^b=de,f^c=fh,g^c=gh,e^d=eh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #153195, #153196, #153197, … |
| maximal normal subgroup of | #153195, #153196, #153197, … |
| maximal quotient of | #345808, #345809, #345810, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24268);