#449111

order	512
nilpotency class	3
isoclinism	isoclinic class of #421447
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2474
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15931
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

720 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)(3,14,19,43)(4,20)(5,11)(6,12)(7,28)(8,13)(9,17,33,18)(10,23)(15,31,21,34)(16,38,46,60)(22,51)(24,36)(25,56,48,32)(26,39)(27,41)(29,37)(30,40)(35,54)(42,47,59,45)(44,61,50,58)(49,55)(52,64,57,62)(53,63);(1,3,7,18,8,19,6,17)(2,9,12,14,13,33,28,43)(4,21,23,47,24,15,29,45)(5,25,26,46,27,48,30,16)(10,34,20,42,37,31,36,59)(11,38,40,32,41,60,39,56)(22,52,54,50,55,57,53,44)(35,62,51,58,63,64,49,61);(1,4,6,23,8,24,7,29)(2,10,12,36,13,37,28,20)(3,15,17,45,19,21,18,47)(5,22,26,53,27,55,30,54)(9,31,14,42,33,34,43,59)(11,35,39,51,41,63,40,49)(16,44,25,57,46,50,48,52)(32,58,38,64,56,61,60,62);(1,5,6,26,8,27,7,30)(2,11,12,39,13,41,28,40)(3,16,18,48,19,46,17,25)(4,22,23,53,24,55,29,54)(9,32,43,60,33,56,14,38)(10,35,36,51,37,63,20,49)(15,44,47,52,21,50,45,57)(31,58,59,62,34,61,42,64)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 356307);