#133945

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	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2297 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

780 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 82 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)(11,20)(12,24,26,17)(13,16)(14,19)(15,22,29,35)(18,27)(21,34)(23,38)(25,32,39,31)(28,40,30,37)(33,36);(9,11)(10,15)(12,22)(13,23)(14,21)(16,30)(17,20)(18,28)(19,29)(24,34)(25,37)(26,35)(27,36)(31,38)(32,33)(39,40);(1,3)(2,5)(4,8)(6,7)(10,16,19,18)(11,21)(15,28,29,30)(17,31,24,32)(20,33,34,38)(22,35)(23,36)(37,40)

Polycyclic presentation:

2^9|a^2=f,c^2=g,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=ci,f^b=fg,f^c=fhi,f^d=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, 41141);