#108731

order	512
nilpotency class	3
isoclinism	isoclinic class of #8760
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#381
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1434
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

477 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^b=fh,e^c=ehi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15927);