#141756

order	512
nilpotency class	4
isoclinism	isoclinic class of #141746
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1972
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#33717	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

404 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 110 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=e,d^2=gi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,c^b=ch,e^b=ei,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72210551, #72210554, #72210562

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48952);