#12979

order	256
nilpotency class	3
isoclinism	isoclinic class of #453
automorphism	Aut(G) = C₄×#63011 (order 1024) · Inn(G) = #120 · 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	#120
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:

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

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,e^b=eh,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×#12979, #494410, #495679, …
maximal normal subgroup of	C₂×#12979, #494410, #495679, …
maximal quotient of	#136575, #136583, #136591, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 5967);