#13126

order	256
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	Aut(G) = C₄×#63011 (order 1024) · Inn(G) = #124 · Out(G) = C₂³×C₄
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#124
commutator	C₂²×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂	#1924
prime radical	C₂×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

150 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,6,12,7,13,30,45,8,14,31,46,5,11,16,33)(3,15,20,47,18,37,48,43,21,35,28,59,19,9,32,36)(4,22,26,55,25,42,49,44,27,40,29,60,24,10,38,41)(17,50,51,62,23,56,57,64,52,34,54,61,53,39,58,63);(1,3,16,48,5,19,31,20,8,21,30,32,7,18,6,28)(2,9,33,47,11,35,46,36,14,37,45,59,13,15,12,43)(4,23,38,54,24,17,29,57,27,53,49,51,25,52,26,58)(10,39,60,62,40,34,44,63,42,56,55,61,22,50,41,64);(1,4,5,24,8,27,7,25)(2,10,11,40,14,42,13,22)(3,17,18,23,21,52,19,53)(6,29,16,49,31,26,30,38)(9,34,15,39,37,50,35,56)(12,44,33,55,46,41,45,60)(20,54,48,58,28,51,32,57)(36,62,47,64,43,61,59,63)

Polycyclic presentation:

2^8|a^2=f,b^2=df,c^2=d,d^2=h,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,c^b=cg,d^b=dh,f^b=fh,e^c=eh,f^c=fh

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#13126, #494840, #496033, …
maximal normal subgroup of	C₂×#13126, #494840, #496033, …
maximal quotient of	#140132, #140142, #140150, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6114);