#13157

order	256
nilpotency class	4
isoclinism	isoclinic class of #13156
automorphism	Aut(G) order 4096
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#1936
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂³	#137
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

148 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 57 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#13157, #494936, #495900, …
maximal normal subgroup of	C₂×#13157, #494936, #495900, …
maximal quotient of	#140189, #140211, #140282, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6145);