#12987

order	256
nilpotency class	3
isoclinism	isoclinic class of #471
automorphism	Aut(G) = C₄×#63011 (order 1024) · Inn(G) = #121 · Out(G) = C₂³×C₄
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#121
commutator	C₂×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂	#1869
prime radical	#575	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

228 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 81 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#12987, #494430, #495804, …
maximal normal subgroup of	C₂×#12987, #494430, #495804, …
maximal quotient of	#136579, #136587, #136595, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 5975);