#142049

order	512
nilpotency class	4
isoclinism	isoclinic class of #125537
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1514
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

352 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 82 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=def,c^2=egi,d^2=ghi,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,h^a=hi,c^b=cg,d^b=dh,f^b=fh,g^b=gi,h^b=hi,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, 49245);