#140680

order	512
nilpotency class	4
isoclinism	isoclinic class of #140671
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1964
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂³×D₈	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

577 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 577 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,c^2=eg,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ch,d^b=dg,e^b=ei,f^b=fg,g^b=gi,d^c=di,f^c=fi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47876);