#134225

order	512
nilpotency class	3
isoclinism	isoclinic class of #8626
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#380
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1797
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

518 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=d,d^2=i,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,c^b=ci,e^c=ei,f^c=fgi

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, 41421);