#108625

order	512
nilpotency class	3
isoclinism	isoclinic class of #8626
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#380
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

448 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gh,e^c=eh,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15821);