#435063

order	512
nilpotency class	3
isoclinism	isoclinic class of #419464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2557
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15514
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

754 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,g^b=gi,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, 342259);