#135677

order	512
nilpotency class	3
isoclinism	isoclinic class of #10370
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#124
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1832
prime radical	C₂³	#424
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

184 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=eg,c^2=d,d^2=gi,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=ch,d^b=di,f^b=fi,e^c=ei,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, 42873);