#133208

order	512
nilpotency class	4
isoclinism	isoclinic class of #133077
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#124 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 859 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) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=i,d^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dh,f^b=fgh,f^c=fh,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40404);