#507918

order	512
nilpotency class	4
isoclinism	isoclinic class of #501524
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 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 398 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)(11,17)(12,27)(13,19)(14,31)(15,30)(16,21)(18,22)(20,29)(23,37)(24,34)(25,40)(26,35)(28,38)(32,39)(33,36);(1,3,4,8)(2,5,6,7)(9,11)(10,17)(12,28)(13,24)(14,32)(15,25)(16,26)(18,33)(19,34)(20,23)(21,35)(22,36)(27,38)(29,37)(30,40)(31,39);(9,12,29,21,15,19,13,30,16,20,27,10,18,31,14,22)(11,23,38,35,25,39,32,40,26,28,37,17,33,34,24,36);(2,6)(5,7)(9,13,15,27,16,14,18,29)(10,19,30,12,21,31,22,20)(11,24,25,38,26,32,33,37)(17,34,40,28,35,39,36,23)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=fh,d^2=gh,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,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, 415114);