#346499

order	512
nilpotency class	3
isoclinism	isoclinic class of #346482
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62873	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2039 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=i,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fh,e^d=eh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253695);