#135547

order	512
nilpotency class	4
isoclinism	isoclinic class of #135547
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8748
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

455 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 455 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 32

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42743);