#475425

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	#17048
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

560 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=eh,c^2=e,d^2=eg,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=de,e^d=ei,f^d=fi,g^d=gi,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, 382621);