#403155

order	512
nilpotency class	3
isoclinism	isoclinic class of #394482
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2435
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14868
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1162 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)(3,15,19,32,20,33,18,9)(4,10,29,36,24,38,23,37)(5,25,26,11,28,40,27,39)(16,42,21,49,44,34,48,30)(17,31,47,57,46,58,45,41)(22,51,53,35,54,55,52,60)(43,56,62,59,63,64,50,61);(1,3,8,18,7,20,6,19)(2,9,12,33,14,32,13,15)(4,21,29,44,24,48,23,16)(5,17,26,47,28,46,27,45)(10,34,36,30,38,42,37,49)(11,31,40,57,39,58,25,41)(22,50,52,63,54,62,53,43)(35,59,51,56,60,61,55,64);(1,4)(2,10)(3,16)(5,22)(6,23)(7,24)(8,29)(9,30)(11,35)(12,36)(13,37)(14,38)(15,34)(17,43)(18,21)(19,48)(20,44)(25,55)(26,52)(27,53)(28,54)(31,56)(32,49)(33,42)(39,60)(40,51)(41,59)(45,62)(46,63)(47,50)(57,61)(58,64);(1,5,8,26,7,28,6,27)(2,11,12,40,14,39,13,25)(3,17,18,47,20,46,19,45)(4,22,29,52,24,54,23,53)(9,31,33,57,32,58,15,41)(10,35,36,51,38,60,37,55)(16,43,21,50,44,63,48,62)(30,56,42,61,49,64,34,59)

Polycyclic presentation:

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

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, 310351);