#415123

order	512
nilpotency class	3
isoclinism	isoclinic class of #401790
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2485
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15183
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

839 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:

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

Polycyclic presentation:

2^9|a^2=fh,c^2=e,d^2=h,e^2=i,g^2=i,h^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,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322319);