#416288

order	512
nilpotency class	3
isoclinism	isoclinic class of #402003
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2494
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15220
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

885 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=eh,d^2=h,f^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,f^b=fi,g^b=gi,e^c=ei,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, 323484);