#507921

order	512
nilpotency class	4
isoclinism	isoclinic class of #501527
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

404 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 120 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 — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=fh,d^2=ghi,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fi,d^c=dh,g^c=gi,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, 415117);