#140518

order	512
nilpotency class	3
isoclinism	isoclinic class of #9755
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×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	#3147	C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 426 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)(3,6)(7,8)(9,10,14,21,16,20,29,35,15,19,22,18,13,30,12,28)(11,17,26,32,27,33,23,37,24,34,31,36,25,39,38,40);(1,2,6,8,4,7,3,5)(9,11,22,26,16,27,12,23,15,24,14,31,13,25,29,38)(10,17,21,36,20,33,35,40,19,34,18,32,30,39,28,37);(9,12,13,22,15,29,16,14)(10,18,20,28,19,21,30,35)(11,23,25,26,24,38,27,31)(17,32,33,37,34,36,39,40)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,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	#72210518
maximal normal subgroup of	#72210518
maximal quotient of	#72210441, #72210451, #72210456

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47714);