#509387

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3061	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

505 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=ei,d^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,g^b=gh,h^b=hi,f^c=fi,g^c=gh,h^c=hi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416583);