#135809

order	512
nilpotency class	4
isoclinism	isoclinic class of #135790
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1846
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#499
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

173 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 80 points:

(1,2,5,8,6,9,4,14)(3,10,13,15,11,16,12,7)(17,18,22,28,23,26,44,29)(19,30,33,53,34,24,35,47)(20,36,38,68,39,67,40,72)(21,41,42,27,43,52,71,51)(25,48,50,37,49,69,54,70)(31,57,59,78,60,66,65,79)(32,61,62,46,63,73,64,56)(45,74,75,80,55,77,76,58);(1,3,4,12,6,11,5,13)(2,7,8,10,9,15,14,16)(17,19,22,35,23,34,44,33)(18,24,28,53,26,30,29,47)(20,31,40,59,39,60,38,65)(21,32,42,64,43,63,71,62)(25,45,54,75,49,55,50,76)(27,46,52,61,51,56,41,73)(36,66,72,79,67,57,68,78)(37,58,48,74,70,80,69,77);(3,11)(7,15)(10,16)(12,13)(17,20,21,37,22,40,42,48,23,39,43,70,44,38,71,69)(18,25,27,36,29,50,41,68,26,49,51,67,28,54,52,72)(19,31,32,58,35,59,64,74,34,60,63,80,33,65,62,77)(24,45,46,66,47,76,73,78,30,55,56,57,53,75,61,79)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43005);