#133903

order	512
nilpotency class	4
isoclinism	isoclinic class of #133903
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, cosocle	C₂²×#41	C₂³
prime radical	#34633	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2297 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

959 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 959 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,c^b=ci,f^b=fgi,f^c=fh,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, 41099);