#474762

order	512
nilpotency class	3
isoclinism	isoclinic class of #467073
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2640
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17006
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gh,c^2=g,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^c=de,e^c=ei,f^c=fi,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, 381958);