#134527

order	512
nilpotency class	4
isoclinism	isoclinic class of #124679
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#410
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
socle	C₂²	#1799
prime radical	#392	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

439 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 439 conjugacy classes of subgroups — too many to draw. Only its 94 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)(3,5)(4,7)(6,8)(9,10)(11,19,20,34)(12,17)(13,25,26,16)(14,18)(15,21,30,35)(22,32)(23,38,39,31)(24,27)(28,40)(29,37)(33,36);(9,11,12,21,14,20,24,35)(10,15,27,34,18,30,17,19)(13,22,23,37,26,36,39,40)(16,29,38,32,25,28,31,33);(2,5)(3,8)(6,7)(10,16)(11,20)(15,28)(17,31)(18,25)(19,32)(21,35)(22,36)(27,38)(29,30)(33,34)(37,40)

Polycyclic presentation:

2^9|a^2=f,b^2=dgh,d^2=i,e^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gi,c^b=ci,f^b=fi,e^c=eh,f^c=fgh,f^e=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41723);