#347169

order	512
nilpotency class	3
isoclinism	isoclinic class of #347154
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62873	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1946 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:

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

Polycyclic presentation:

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

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, 254365);