#427166

order	512
nilpotency class	3
isoclinism	isoclinic class of #418498
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2673
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15366
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

749 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)(3,15,19,34)(4,10)(5,25)(6,12)(7,13)(8,14)(9,18,33,29)(11,27)(16,43,46,35)(17,47,48,32)(20,50,45,31)(21,52)(22,37)(23,42)(24,38)(26,56,49,39)(28,40)(30,41)(36,53)(44,58,64,60)(51,63,57,59)(54,62)(55,61);(1,3,6,29,8,19,7,18)(2,9,12,34,14,33,13,15)(4,20,22,46,24,45,23,16)(5,26,27,17,28,49,30,48)(10,35,37,31,38,43,42,50)(11,39,40,32,41,56,25,47)(21,44,53,51,55,64,54,57)(36,60,61,59,62,58,52,63);(1,4,6,22,8,24,7,23)(2,10,12,37,14,38,13,42)(3,16,29,20,19,46,18,45)(5,21,27,53,28,55,30,54)(9,31,34,43,33,50,15,35)(11,36,40,61,41,62,25,52)(17,44,49,51,48,64,26,57)(32,58,56,63,47,60,39,59);(1,5,6,27,8,28,7,30)(2,11,12,40,14,41,13,25)(3,17,18,26,19,48,29,49)(4,21,22,53,24,55,23,54)(9,32,15,39,33,47,34,56)(10,36,37,61,38,62,42,52)(16,44,45,57,46,64,20,51)(31,58,35,59,50,60,43,63)

Polycyclic presentation:

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

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, 334362);