#132384

order	512
nilpotency class	4
isoclinism	isoclinic class of #132181
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	C₂×#546	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#546 ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

834 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 834 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,f^c=fgi,f^d=fi,f^e=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, 39580);