#104316

order	512
nilpotency class	3
isoclinism	isoclinic class of #104309
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

647 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 647 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=gi,d^2=g,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fgi,c^b=cf,d^b=dg,e^b=eghi,f^b=fg,d^c=dh,e^c=ei,f^c=fg

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 11512);