#108868

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#117
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₄⋊#95
prime radical	C₂⁴	C₄×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×#431 ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

608 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,4,7,6,8,5,3)(9,10)(11,20,23,16)(12,25)(13,29)(14,19)(15,18)(17,26)(21,35,37,39)(22,31,24,34)(27,32)(28,40)(30,36,33,38);(1,3,5,8,6,7,4,2)(9,11,14,24,15,23,13,22)(10,16,19,31,18,20,29,34)(12,21,27,38,28,37,26,36)(17,30,25,39,32,33,40,35);(9,12,13,26,15,28,14,27)(10,17,19,25,18,32,29,40)(11,21,22,36,23,37,24,38)(16,30,31,39,20,33,34,35)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 16064);