#140195

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 86 normal subgroups are shown.

normal subgroups only

This group has 253 conjugacy classes of subgroups — too many to draw. Only its 86 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)(9,10,13,19,15,20,30,22,14,18,21,26,31,35,36,38)(11,16,24,29,25,32,39,40,23,12,17,27,33,28,34,37);(1,2,5,7,6,8,4,3)(9,11,21,24,15,25,36,39,14,23,13,17,31,33,30,34)(10,16,19,27,20,32,22,37,18,12,26,29,35,28,38,40);(1,3,4,8,6,7,5,2)(9,12)(10,17,18,24)(11,22,23,38)(13,29)(14,16)(15,28)(19,25,26,33)(20,34,35,39)(21,27)(30,40)(31,32)(36,37)

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=ch,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, 47391);