#474436

order	512
nilpotency class	3
isoclinism	isoclinic class of #460462
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2499
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16965
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

562 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=eg,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,d^c=de,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, 381632);