#140207

order	512
nilpotency class	3
isoclinism	isoclinic class of #10370
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#124
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1924
prime radical	#532	C₂×C₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

253 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 253 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=d,d^2=gi,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=cgh,d^b=di,f^b=fi,e^c=ei,f^c=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, 47403);