#13199

order	256
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	Aut(G) = C₄×#511819 (order 2048) · Inn(G) = #124 · Out(G) = C₂×C₄×D₄
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

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	#1955
prime radical	#575	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

203 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 203 conjugacy classes of subgroups — too many to draw. Only its 69 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#143173, #143174, #143175, …
maximal normal subgroup of	#143173, #143174, #143175, …
maximal quotient of	#140562, #140566, #140584, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6187);