#108908

order	512
nilpotency class	3
isoclinism	isoclinic class of #108898
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₄⋊#95
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#124
LCS	G ▷ C₄² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

530 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ei,d^2=hi,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=dh,f^b=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, 16104);