#471103

order	512
nilpotency class	3
isoclinism	isoclinic class of #462392
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2517
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16752
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

518 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)(3,14)(4,20,25,53)(5,26,30,11)(6,31)(7,12)(8,13)(9,17)(10,24,36,23)(15,43,21,32)(16,42,50,60)(18,34)(19,39)(22,37)(27,35)(28,38,29,40)(33,48,57,49)(41,46,51,45)(44,63)(47,59)(52,56)(54,58)(55,61)(62,64);(1,3,7,17,8,19,6,18)(2,9,12,39,13,34,31,14)(4,21,24,46,25,15,23,45)(5,16,28,49,30,50,29,48)(10,32,20,51,36,43,53,41)(11,33,40,60,26,57,38,42)(22,54,27,47,56,44,55,64)(35,58,52,59,61,63,37,62);(1,4,6,23,8,25,7,24)(2,10,12,20,13,36,31,53)(3,15,17,45,19,21,18,46)(5,27,28,56,30,55,29,22)(9,32,14,41,34,43,39,51)(11,37,38,61,26,52,40,35)(16,47,48,54,50,64,49,44)(33,59,60,63,57,62,42,58);(1,5)(2,11,13,26)(3,16)(4,22)(6,29)(7,28)(8,30)(9,33,34,57)(10,35,36,61)(12,40,31,38)(14,42,39,60)(15,44)(17,49)(18,48)(19,50)(20,52,53,37)(21,54)(23,55)(24,27)(25,56)(32,58,43,63)(41,62,51,59)(45,64)(46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378299);