#140504

order	512
nilpotency class	4
isoclinism	isoclinic class of #140463
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1948
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#475
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

188 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,5,10,6,11,15,9)(3,7,4,13,12,8,14,16)(17,18)(19,27,29,43)(20,31)(21,24)(22,36)(23,28,38,44)(25,34)(26,35)(30,45,37,41)(32,39,47,42)(33,46)(40,48);(1,3,5,4,6,12,15,14)(2,7,10,13,11,8,9,16)(17,19,22,37,21,29,35,30)(18,23,26,42,24,38,36,39)(20,32,34,28,33,47,48,44)(25,41,31,27,40,45,46,43);(1,4,5,12,6,14,15,3)(2,8,9,13,11,7,10,16)(18,24)(23,38)(25,40)(26,36)(27,43)(31,46)(39,42)(41,45)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287733, #72287756, #72287758
maximal normal subgroup of	#72287733, #72287756, #72287758

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47700);