#507685

order	512
nilpotency class	4
isoclinism	isoclinic class of #501332
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

542 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 130 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 — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=egi,c^2=g,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,c^b=cg,d^b=di,f^c=fh,h^c=hi,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414881);