#133441

order	512
nilpotency class	4
isoclinism	isoclinic class of #133434
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	C₂³×C₄	C₂²×C₄
Frattini, prime radical, cosocle	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #2308 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 702 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)(11,19,22,32)(12,16,24,31)(13,25,27,17)(14,18)(15,21)(20,33)(23,36)(26,29)(28,37,39,34)(30,38)(35,40);(1,2)(3,5)(4,6)(7,8)(9,11,14,22)(10,15,18,29)(12,20,24,35)(13,26,27,21)(16,28,31,39)(17,32,25,19)(23,37,38,34)(30,40,36,33);(1,3)(2,5)(4,7)(6,8)(9,12)(10,16,18,31)(11,20)(13,23)(14,24)(15,28,29,39)(17,30,25,36)(19,33,32,40)(21,34)(22,35)(26,37)(27,38)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=di,e^b=eh,f^b=fgi,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	#72303830, #72303832, #72303860
maximal normal subgroup of	#72303830, #72303832, #72303860
maximal quotient of	#72303937

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40637);