#13146

order	256
nilpotency class	4
isoclinism	isoclinic class of #2221
automorphism	Aut(G) = #72312248 (order 2048) · Inn(G) = #466 · Out(G) = C₂³×C₄
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#466
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂	#1934
prime radical	C₂×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 156 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:

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

Polycyclic presentation:

2^8|a^2=f,b^2=dfg,c^2=d,d^2=gh,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,e^c=eh

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#13146, #494901, #495893, …
maximal normal subgroup of	C₂×#13146, #494901, #495893, …
maximal quotient of	#140172, #140173, #140197, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6134);