#135546

order	512
nilpotency class	4
isoclinism	isoclinic class of #135543
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	C₂³×Q₈	C₂³
prime radical	C₂⁴	#117
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 435 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=d,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,f^b=fgi,e^c=eg,f^c=fh,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, 42742);