Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | Aut(G) = #72234841 (order 2048) · Inn(G) = B(4,𝔽2) · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C24 |
| socle | C2 | C2×B(4,𝔽2) |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
572 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 572 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)(7,14)(12,23)(13,24)(15,20)(16,21)(17,22)(18,26)(19,27)(25,31)(28,30)(29,32);(1,3)(2,7)(4,13)(5,14)(6,12)(8,19)(9,11)(10,18)(15,27)(16,25)(17,26)(20,24)(21,29)(22,23)(28,32)(30,31);(3,12)(5,17)(7,18)(11,23)(13,25)(15,28)(19,29)(24,31);(1,4,6,16)(2,8,10,21)(3,13,12,25)(5,15,17,28)(7,19,18,29)(9,20,22,30)(11,24,23,31)(14,27,26,32)Polycyclic presentation:
2^8|d^2=h,b^a=be,c^a=cf,g^a=gh,c^b=ch,f^b=fgh,e^c=eg,f^e=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #152560, #153131, #153171, … |
| maximal normal subgroup of | #152560, #153131, #153171, … |
| maximal quotient of | #484156, #484175, #484242, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26538);