#140512

order	512
nilpotency class	3
isoclinism	isoclinic class of #9755
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1954
prime radical	C₂³	#408
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

340 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 130 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)(3,7)(6,8)(9,10,14,18,16,20,12,26,15,19,28,31,13,29,27,39)(11,17,24,34,25,33,37,40,22,35,30,32,23,38,21,36);(1,2,3,5,4,6,7,8)(9,11,14,30,16,25,12,21,15,22,28,24,13,23,27,37)(10,17,31,34,20,33,39,40,19,35,18,32,29,38,26,36);(9,12,13,14,15,27,16,28)(10,18,20,26,19,31,29,39)(11,21,23,30,22,37,25,24)(17,32,33,36,35,34,38,40)

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=e,d^2=gi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=di,f^b=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210514
maximal normal subgroup of	#72210514
maximal quotient of	#72210441

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47708);