#133846

order	512
nilpotency class	4
isoclinism	isoclinic class of #133843
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	#35149	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#536 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

1512 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 1512 conjugacy classes of subgroups — too many to draw. Only its 90 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)(12,23)(13,26,28,18)(14,29)(15,20)(16,22,32,27)(17,21)(24,31,38,37)(25,35)(30,34)(33,36,39,40);(9,11,15,14)(10,16,20,32)(12,21,25,30)(13,22,28,27)(17,31,34,37)(18,19,26,29)(23,33,35,39)(24,36,38,40);(1,3)(2,5)(4,8)(6,7)(9,12,15,25)(10,17,20,34)(11,21,14,30)(13,24,28,38)(16,31,32,37)(18,33,26,39)(19,35,29,23)(22,36,27,40)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,f^b=fg,e^c=ei,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, 41042);