#132328

order	512
nilpotency class	4
isoclinism	isoclinic class of #132309
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2353 ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

504 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=h,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,d^b=dh,e^b=ehi,f^b=fgh,d^c=dhi,f^c=fg,f^d=fi,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 39524);