#134401

order	512
nilpotency class	3
isoclinism	isoclinic class of #8674
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#380
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1797
prime radical	C₂³	#407
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

338 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 118 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dg,c^2=g,d^2=i,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,e^c=egi,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41597);