#498291

order	512
nilpotency class	3
isoclinism	isoclinic class of #3290
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	D₄²
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#22985
prime radical	#3067	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

606 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=e,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,d^b=de,e^b=ei,g^b=gi,f^c=fi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405487);