#485026

order	512
nilpotency class	4
isoclinism	isoclinic class of #484212
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3203	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2922 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

683 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 80 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=h,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fi,g^a=gi,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 392222);