#449153

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2474
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15933
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

573 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)(3,14,18,32)(4,19)(5,10)(6,11)(7,13)(8,17,33,46)(9,23)(12,29)(15,34,20,30)(16,37,48,60)(21,52)(22,36)(24,40)(25,57,47,31)(26,38)(27,41)(28,39)(35,54)(42,44,49,45)(43,59,51,61)(50,55)(53,62,58,64)(56,63);(1,3,7,18)(2,8)(4,20)(5,25)(6,17,29,46)(9,34,36,30)(10,37,39,60)(11,14)(12,32)(13,33)(15,24)(16,27)(19,49,40,42)(21,53,55,58)(22,45)(23,44)(26,48)(28,47)(31,38,57,41)(35,62)(43,56,51,54)(50,59)(52,61)(63,64);(1,4,6,22,7,24,29,23)(2,9,11,19,13,36,12,40)(3,15,17,44,18,20,46,45)(5,21,26,54,28,55,27,56)(8,30,14,42,33,34,32,49)(10,35,38,50,39,63,41,52)(16,43,25,58,48,51,47,53)(31,59,37,64,57,61,60,62);(1,5,6,26,7,28,29,27)(2,10,11,38,13,39,12,41)(3,16,46,47,18,48,17,25)(4,21,22,54,24,55,23,56)(8,31,32,60,33,57,14,37)(9,35,19,50,36,63,40,52)(15,43,45,53,20,51,44,58)(30,59,49,62,34,61,42,64)

Polycyclic presentation:

2^9|a^2=g,b^2=ef,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, 356349);