#134604

order	512
nilpotency class	4
isoclinism	isoclinic class of #134444
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

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	#3147	C₂²
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

376 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 102 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dg,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, 41800);