#132232

order	512
nilpotency class	4
isoclinism	isoclinic class of #132201
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	C₂×#553	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#553 ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 676 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)(11,20,24,33)(12,25)(13,29,14,18)(15,19)(16,22)(17,27,32,26)(21,30,38,40)(23,31)(28,34)(35,36)(37,39);(1,2)(3,7)(4,6)(5,8)(9,11,15,24)(10,16,19,31)(12,21,28,38)(13,23,14,22)(17,30,32,40)(18,33,29,20)(25,35,34,39)(26,37,27,36);(3,8)(5,7)(9,12,15,28)(10,17,19,32)(11,21,24,38)(13,27,14,26)(16,30,31,40)(18,34,29,25)(20,35,33,39)(22,37,23,36)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,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=eh,f^b=fgh,d^c=dh,e^c=ei,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, 39428);