#133500

order	512
nilpotency class	4
isoclinism	isoclinic class of #133350
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂²×D₄	C₂²×C₄
Frattini, prime radical, cosocle	C₂³×D₄	C₂³
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

690 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340902, #72341181, #72341184
maximal normal subgroup of	#72340902, #72341181, #72341184

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40696);