#509776

order	512
nilpotency class	4
isoclinism	isoclinic class of #509163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33527	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

672 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)(3,7)(4,6)(5,8)(9,10,13,20,16,23,15,22)(11,17,25,19,27,42,34,49)(12,28,30,47,33,46,32,18)(14,21,35,50,39,53,38,52)(24,54,55,61,56,44,57,40)(26,41,36,48,59,68,63,69)(29,60,58,70,64,71,67,43)(31,62,65,72,66,51,37,45);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,29)(13,25)(14,36)(15,34)(16,27)(18,43)(19,20)(21,48)(22,49)(23,42)(24,31)(26,38)(28,60)(30,58)(32,67)(33,64)(35,59)(37,57)(39,63)(40,45)(41,52)(44,51)(46,71)(47,70)(50,68)(53,69)(54,62)(55,65)(56,66)(61,72);(9,12,16,33)(10,18,23,47)(11,24,27,56)(13,32,15,30)(14,37,39,65)(17,40,42,61)(19,44,49,54)(20,46,22,28)(21,51,53,62)(25,57,34,55)(26,58,59,67)(29,63,64,36)(31,35,66,38)(41,60,68,71)(43,69,70,48)(45,50,72,52);(2,6)(3,8)(9,11)(10,19)(12,24)(13,34)(14,26)(15,25)(16,27)(17,20)(18,44)(21,48)(22,42)(23,49)(28,61)(29,31)(30,57)(32,55)(33,56)(35,63)(36,38)(37,58)(39,59)(40,46)(41,50)(43,51)(45,71)(47,54)(52,68)(53,69)(60,72)(62,70)(64,66)(65,67)

Polycyclic presentation:

2^9|a^2=f,c^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defi,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416972);