#129074

order	512
nilpotency class	3
isoclinism	isoclinic class of #8603
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#117
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1770
prime radical	C₂³	#405
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

250 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=de,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ch,f^a=fi,c^b=cg,d^b=di,f^c=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36270);