#472910

order	512
nilpotency class	3
isoclinism	isoclinic class of #455690
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2825
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16841
prime radical	#23412	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1006 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)(3,8)(4,20)(5,10)(6,12)(7,13)(9,23)(11,14)(15,34)(16,46)(17,42)(18,32)(19,33)(21,51)(22,54)(24,36)(25,52)(26,39)(27,41)(28,40)(29,37)(30,44)(31,48)(35,55)(38,56)(43,63)(45,62)(47,64)(49,60)(50,61)(53,59)(57,58);(1,3,14,18,7,19,6,17)(2,8,11,32,13,33,12,42)(4,21,23,15,24,44,29,45)(5,16,27,49,26,48,28,50)(9,34,36,30,37,62,20,51)(10,31,41,61,39,46,40,60)(22,53,56,43,25,57,55,47)(35,63,54,58,38,64,52,59);(1,4)(2,9)(3,15)(5,25)(6,29)(7,24)(8,30)(10,38)(11,36)(12,20)(13,37)(14,23)(16,47)(17,21)(18,44)(19,45)(22,26)(27,55)(28,56)(31,59)(32,62)(33,51)(34,42)(35,39)(40,54)(41,52)(43,48)(46,58)(49,53)(50,57)(60,64)(61,63);(1,5,6,28,7,26,14,27)(2,10,11,41,13,39,12,40)(3,16,17,50,19,48,18,49)(4,22,23,56,24,25,29,55)(8,31,32,61,33,46,42,60)(9,35,20,52,37,38,36,54)(15,43,44,57,45,47,21,53)(30,58,34,63,51,59,62,64)

Polycyclic presentation:

2^9|b^2=egh,d^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,g^b=gi,h^b=hi,d^c=de,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi

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, 380106);