#474379

order	512
nilpotency class	3
isoclinism	isoclinic class of #460311
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2500
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16962
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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=g,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^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, 381575);