#507768

order	512
nilpotency class	4
isoclinism	isoclinic class of #500596
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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

450 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 122 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=e,d^2=gh,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,f^b=fi,e^c=ei,f^c=fh,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414964);