#504884

order	512
nilpotency class	4
isoclinism	isoclinic class of #500857
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

561 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)(11,24)(12,30)(13,36)(14,41)(15,22)(16,23)(17,46)(18,27)(19,34)(20,26)(21,43)(25,58)(28,55)(29,60)(31,57)(32,52)(33,59)(35,54)(37,51)(38,66)(39,71)(40,48)(42,47)(44,61)(45,49)(50,65)(53,62)(56,63)(64,70)(67,69)(68,72);(1,3,4,8)(2,5,6,7)(9,11,25,32,17,29,14,27,15,28,43,37,16,44,42,65)(10,18,47,60,23,52,21,24,22,50,41,61,46,51,58,55)(12,31,62,40,35,26,33,38,34,64,69,39,45,63,68,13)(19,36,59,56,54,71,53,70,30,66,72,20,49,48,67,57);(2,6)(5,7)(9,12)(10,19)(11,26)(13,37)(14,33)(15,34)(16,45)(17,35)(18,48)(20,55)(21,53)(22,30)(23,54)(24,56)(25,62)(27,39)(28,63)(29,64)(31,44)(32,38)(36,52)(40,65)(41,72)(42,68)(43,69)(46,49)(47,59)(50,71)(51,66)(57,60)(58,67)(61,70);(2,6)(5,7)(9,13,25,31,17,40,14,26,15,38,43,64,16,39,42,63)(10,20,47,48,23,57,21,36,22,56,41,71,46,70,58,66)(11,12,32,62,29,35,27,33,28,34,37,69,44,45,65,68)(18,49,60,67,52,19,24,59,50,54,61,53,51,30,55,72)

Polycyclic presentation:

2^9|b^2=ehi,d^2=efh,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412080);