#452649

order	512
nilpotency class	3
isoclinism	isoclinic class of #427311
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2471
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16057
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

756 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 359845);