#132331

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×Q₈	C₂³
prime radical	C₂³	#379
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2356 ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 468 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=h,e^2=i,f^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, 39527);