#140205

order	512
nilpotency class	3
isoclinism	isoclinic class of #10370
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂

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	C₂³	#435
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 226 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,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, 47401);