#459715

order	512
nilpotency class	3
isoclinism	isoclinic class of #459714
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16347
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

834 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 366911);