#471151

order	512
nilpotency class	3
isoclinism	isoclinic class of #459756
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2761
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16763
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

515 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:

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

Polycyclic presentation:

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

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, 378347);