#330989

order	512
nilpotency class	3
isoclinism	isoclinic class of #330975
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

496 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=di,f^a=fh,c^b=cg,e^b=eh,g^b=gi,d^c=de,f^c=fh,g^c=ghi,g^d=gh

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238185);