#509781

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	#25799	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1757 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

659 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)(3,14)(4,21,24,10)(5,26)(6,12)(7,11)(8,13)(9,17)(15,18)(16,36,46,42)(19,35)(20,40)(22,33,48,60)(23,37,27,39)(25,51,55,38)(28,58)(29,44)(30,31)(32,41)(34,49)(43,47,61,56)(45,50)(52,64,63,53)(54,62,59,57);(1,3,8,20)(2,9,13,35)(4,22,24,48)(5,19,7,17)(6,29,32,49)(10,36,21,42)(11,40,26,14)(12,15,41,45)(16,27,46,23)(18,28,50,31)(25,56,55,47)(30,34,58,44)(33,39,60,37)(38,52,51,63)(43,57,61,62)(53,59,64,54);(1,2)(3,15)(4,10)(5,11)(6,30)(7,26)(8,13)(9,29)(12,31)(14,18)(16,43)(17,44)(19,34)(20,45)(21,24)(22,52)(23,37)(25,57)(27,39)(28,41)(32,58)(33,53)(35,49)(36,56)(38,54)(40,50)(42,47)(46,61)(48,63)(51,59)(55,62)(60,64);(1,4)(2,10)(3,16)(5,27)(6,25)(7,23)(8,24)(9,33)(11,39)(12,38)(13,21)(14,42)(15,43)(17,48)(18,47)(19,22)(20,46)(26,37)(28,59)(29,53)(30,57)(31,54)(32,55)(34,52)(35,60)(36,40)(41,51)(44,63)(45,61)(49,64)(50,56)(58,62)

Polycyclic presentation:

2^9|a^2=f,b^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, 416977);