Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33568 |
| automorphism | Aut(G) = #72256403 (order 2048) · Inn(G) = C2×B(4,𝔽2) · Out(G) = C24 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C24 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×Q8 ◁ G | |
| LECS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
555 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 555 conjugacy classes of subgroups — too many to draw. Only its 109 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)(3,12)(4,16)(5,8)(6,11)(7,14)(9,19)(10,21)(13,27)(15,29)(17,24)(18,26)(20,30)(22,28)(23,31)(25,32);(1,3)(2,7)(4,17)(5,20)(6,13)(8,23)(9,27)(10,25)(11,24)(12,16)(14,19)(15,21)(18,31)(22,32)(26,29)(28,30);(2,8)(5,21)(7,23)(9,28)(10,11)(12,29)(15,20)(16,26)(18,22)(24,25)(27,30)(31,32);(3,13)(7,24)(12,27)(14,17)(15,20)(23,25)(29,30)(31,32)Polycyclic presentation:
2^8|e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gh,d^b=dh,e^b=eh,f^b=fgh,e^c=eg,f^c=fh,f^e=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #152569, #153132, #153169, … |
| maximal normal subgroup of | #152569, #153132, #153169, … |
| maximal quotient of | #484176, #484181, #484262, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26556);