#133489

order	512
nilpotency class	4
isoclinism	isoclinic class of #133422
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂⁵	C₂²×C₄
Frattini, prime radical, cosocle	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #2308 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 694 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)(11,19)(12,16,24,30)(13,25,26,17)(14,18)(15,21,28,35)(20,31,34,40)(22,32)(23,36)(27,38)(29,37)(33,39);(1,2)(3,5)(4,6)(7,8)(9,11,14,22)(10,15,18,28)(12,20,24,34)(13,21,26,35)(16,27,30,39)(17,19,25,32)(23,33,37,38)(29,31,36,40);(1,3)(2,5)(4,7)(6,8)(9,12,14,24)(10,16)(11,20,22,34)(13,23,26,37)(15,27)(17,29)(18,30)(19,31)(21,33,35,38)(25,36)(28,39)(32,40)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303838, #72303843, #72303855
maximal normal subgroup of	#72303838, #72303843, #72303855
maximal quotient of	#72303978

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40685);