#140694

order	512
nilpotency class	4
isoclinism	isoclinic class of #140671
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

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	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 375 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) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=egi,d^2=i,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=fgi,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, 47890);