#448850

order	512
nilpotency class	3
isoclinism	isoclinic class of #425083
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2576
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15922
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

739 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,d^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,g^b=gi,h^b=hi,f^c=fi,h^c=hi,e^d=ei,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, 356046);