#140960

order	512
nilpotency class	4
isoclinism	isoclinic class of #140953
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1964
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 365 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=egi,d^2=gh,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,f^b=fh,g^b=gi,h^b=hi,f^c=fi,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48156);