#140545

order	512
nilpotency class	3
isoclinism	isoclinic class of #9758
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	#532	C₂×C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

327 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 327 conjugacy classes of subgroups — too many to draw. Only its 102 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)(9,10,15,18,16,19,12,28,13,20,22,36,14,31,30,39)(11,21,25,35,27,37,23,38,26,17,32,33,24,34,29,40);(1,2,3,5,4,6,7,8)(9,11,22,25,16,27,30,23,13,26,15,32,14,24,12,29)(10,17,18,35,19,34,28,38,20,21,36,33,31,37,39,40);(2,6)(5,8)(9,12,14,15,13,30,16,22)(10,18,19,28,20,36,31,39)(11,23,24,25,26,29,27,32)(17,33,34,40,21,35,37,38)

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,c^b=cg,d^b=di,f^b=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47741);