#448776

order	512
nilpotency class	3
isoclinism	isoclinic class of #448775
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15919
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

673 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ef,c^2=e,d^2=eg,e^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,f^c=fi,h^c=hi,f^d=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 355972);