#488623

order	512
nilpotency class	4
isoclinism	isoclinic class of #488539
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

693 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:

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

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=dh,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, 395819);