#507919

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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 415 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)(14,30)(15,22)(16,29)(18,21)(20,28)(23,37)(24,32)(25,34)(26,38)(31,40)(33,35)(36,39);(1,3,4,8)(2,5,6,7)(9,11)(10,17)(12,23)(13,24)(14,31)(15,25)(16,26)(18,35)(19,32)(20,36)(21,33)(22,34)(27,37)(28,39)(29,38)(30,40);(9,12,28,22,16,19,13,29,15,20,27,10,18,30,14,21)(11,23,37,17,26,32,31,33,25,36,39,34,35,40,24,38);(2,6)(5,7)(9,13,16,27,15,14,18,28)(10,19,29,12,22,30,21,20)(11,24,26,37,25,31,35,39)(17,32,38,23,34,40,33,36)

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,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, 415115);