#479844

order	512
nilpotency class	3
isoclinism	isoclinic class of #477835
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2793
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17276
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

502 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)(3,15)(4,21)(5,11,29,41)(6,13)(7,12)(8,14)(9,18)(10,24)(16,35)(17,32,49,44)(19,33)(20,34)(22,31)(23,51,54,63)(25,38)(26,37)(27,56,50,39)(28,42,30,40)(36,55,62,53)(43,47)(45,64,48,61)(46,59)(52,60,57,58);(1,3)(2,9,14,15)(4,22)(5,27,29,50)(6,20)(7,19)(8,18)(10,35,38,59)(11,39)(12,34,13,33)(16,25)(17,28,49,30)(21,43,37,31)(23,48,54,45)(24,46)(26,47)(32,42)(36,60)(40,44)(41,56)(51,64)(52,53,57,55)(58,62)(61,63);(1,4,6,24,8,26,7,25)(2,10,12,21,14,38,13,37)(3,16,19,47,18,46,20,22)(5,23,28,53,29,54,30,55)(9,31,33,59,15,43,34,35)(11,36,40,51,41,62,42,63)(17,48,50,52,49,45,27,57)(32,60,56,61,44,58,39,64);(1,5,6,28,8,29,7,30)(2,11,12,40,14,41,13,42)(3,17,19,50,18,49,20,27)(4,23,24,53,26,54,25,55)(9,32,33,56,15,44,34,39)(10,36,21,51,38,62,37,63)(16,45,47,57,46,48,22,52)(31,58,59,64,43,60,35,61)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387040);