#128222

order	512
nilpotency class	3
isoclinism	isoclinic class of #8603
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²

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	#1746
prime radical	C₂³	#405
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

214 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)(3,6)(4,8)(5,7)(9,13)(10,12)(11,14)(15,16)(17,18,22,28,23,27,45,56,24,29,46,58,21,41,43,71)(19,30,33,60,32,25,47,49,34,50,64,77,42,51,72,78)(20,35,38,66,40,55,70,57,39,67,44,73,37,26,52,54)(31,48,61,75,36,53,68,79,62,65,63,76,69,59,74,80);(1,3,4,9,5,10,11,15)(2,6,8,13,7,12,14,16)(17,19)(18,25)(20,36,39,69)(21,42)(22,33)(23,32)(24,34)(26,53,55,59)(27,50)(28,49)(29,51)(30,41)(31,40,62,37)(35,65,67,48)(38,68,44,74)(43,72)(45,47)(46,64)(52,61,70,63)(54,79,57,80)(56,77)(58,78)(60,71)(66,76,73,75);(2,7)(6,12)(8,14)(13,16)(17,20,21,37,24,39,23,40)(18,26,27,35,29,55,41,67)(19,31,32,36,34,62,42,69)(22,44,43,70,46,38,45,52)(25,48,30,59,51,65,50,53)(28,57,56,73,58,54,71,66)(33,63,47,74,64,61,72,68)(49,76,60,79,78,75,77,80)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=d,d^2=i,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=cgh,d^a=di,c^b=cg,d^b=di,e^c=ei,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, 35418);