Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33557 |
| 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×D4 | C24 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×D4 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
579 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 579 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)(12,24)(13,25)(15,21)(16,22)(17,30)(18,23)(19,26)(20,28)(27,32)(29,31);(1,3,6,13)(2,7,10,20)(4,12,15,27)(5,14,18,28)(8,19,21,31)(9,11,23,25)(16,26,17,29)(22,24,30,32);(1,4)(2,8)(3,12)(5,17)(6,15)(7,19)(9,22)(10,21)(11,24)(13,27)(14,29)(16,18)(20,31)(23,30)(25,32)(26,28);(3,13)(4,15)(7,20)(8,21)(11,25)(14,28)(16,17)(22,30)Polycyclic presentation:
2^8|b^2=h,b^a=be,c^a=cf,g^a=gh,d^b=dh,f^b=fgh,d^c=dh,e^c=eg,f^e=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #152568, #153132, #153167, … |
| maximal normal subgroup of | #152568, #153132, #153167, … |
| maximal quotient of | #484163, #484177, #484249, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26546);