Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33551 |
| automorphism | Aut(G) = C2×#72236702 (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×D4 | C24 |
| prime radical | #3025 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
556 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 556 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)(3,11)(4,8)(5,9)(6,10)(7,13)(12,25)(14,26)(15,29)(16,22)(17,23)(18,24)(19,27)(20,28)(21,30)(31,32);(1,3,6,14)(2,7,10,20)(4,15,17,12)(5,13,18,28)(8,21,23,19)(9,11,24,26)(16,30,31,27)(22,29,32,25);(5,18)(13,28)(16,31)(27,30);(1,4,6,17)(2,8,10,23)(3,12,14,15)(5,16,18,31)(7,19,20,21)(9,22,24,32)(11,25,26,29)(13,27,28,30)Polycyclic presentation:
2^8|b^2=h,d^2=h,b^a=be,c^a=cf,g^a=gh,d^b=dh,f^b=fgh,e^c=eg,f^e=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #499821, #499822, #499829, … |
| maximal normal subgroup of | #499821, #499822, #499829, … |
| maximal quotient of | #484160, #484166, #484185, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26542);