#509780

order	512
nilpotency class	4
isoclinism	isoclinic class of #509779
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#31000
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33541	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1757 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

654 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416976);