#134593

order	512
nilpotency class	4
isoclinism	isoclinic class of #134444
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1802
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#417
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

300 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 114 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=di,c^2=g,d^2=i,e^2=hi,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,e^c=eh,f^c=fgh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41789);