#140361

order	512
nilpotency class	4
isoclinism	isoclinic class of #13459
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#498
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9597
prime radical	C₂³	#444
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

203 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 203 conjugacy classes of subgroups — too many to draw. Only its 92 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=e,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,c^b=ch,e^b=ei,d^c=di,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224951, #72224960, #72224964, …
maximal normal subgroup of	#72224951, #72224960, #72224964, …
maximal quotient of	C₂×#140361

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47557);