#140567

order	512
nilpotency class	3
isoclinism	isoclinic class of #111253
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1955
prime radical	#2397	C₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

418 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 106 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:

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

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,c^b=ch,d^b=di,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47763);