#132730

order	512
nilpotency class	4
isoclinism	isoclinic class of #132541
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂×C₂²⋊C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	C₂×#120	D₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

558 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=hi,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,d^b=dh,e^b=egh,f^b=fghi,d^c=dghi,f^c=fgi,e^d=ei,f^d=fi,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 39926);