Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12155 |
| automorphism | Aut(G) = C25×B(4,𝔽2) (order 2048) · Inn(G) = B(4,𝔽2) · Out(G) = C25 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C4⋊C4 | C23 |
| prime radical | #537 | C22 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
190 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 190 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,6)(3,4)(7,8)(9,10)(11,15)(12,22)(13,26)(14,18)(16,25)(17,27)(19,34)(20,33)(21,30)(23,38)(24,37)(28,35)(29,36)(31,39)(32,40);(1,2)(3,6)(4,8)(5,7)(9,11,14,21)(10,15,18,30)(12,23,24,19)(13,20,27,36)(16,31,32,28)(17,33,26,29)(22,34,37,38)(25,39,40,35);(1,3,5,4)(2,6,7,8)(9,12)(10,16,18,32)(11,19)(13,25)(14,24)(15,28,30,31)(17,22,26,37)(20,35)(21,23)(27,40)(29,34,33,38)(36,39)Polycyclic presentation:
2^8|b^2=h,c^2=fgh,d^2=f,e^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,g^a=gh,c^b=ch,d^b=df,e^b=egh,d^c=dg,e^d=eh25 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12201, #484799, #484801, … |
| maximal normal subgroup of | C2×#12201, #484799, #484801, … |
| maximal quotient of | #125706, #125788, #125821, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5189);