#458077

order	512
nilpotency class	3
isoclinism	isoclinic class of #458076
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16309
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

897 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)(3,15,19,42)(4,10,24,37)(5,11,27,40)(6,14,8,12)(9,29,32,18)(16,43,46,33)(17,47,51,44)(20,53,52,30)(21,34,55,62)(22,36,23,35)(25,57,56,38)(26,41,28,39)(31,50,60,49)(45,61,64,59)(48,63,54,58);(1,3,8,29,7,19,6,18)(2,9,14,42,13,32,12,15)(4,20,22,46,24,52,23,16)(5,17,26,49,27,51,28,50)(10,33,35,53,37,43,36,30)(11,31,39,47,40,60,41,44)(21,54,56,64,55,48,25,45)(34,61,57,63,62,59,38,58);(1,4,8,22,7,24,6,23)(2,10,14,35,13,37,12,36)(3,16,29,20,19,46,18,52)(5,25,26,21,27,56,28,55)(9,30,42,33,32,53,15,43)(11,38,39,34,40,57,41,62)(17,48,49,45,51,54,50,64)(31,59,47,58,60,61,44,63);(1,5)(2,11)(3,17)(4,21)(6,28)(7,27)(8,26)(9,31)(10,34)(12,41)(13,40)(14,39)(15,44)(16,45)(18,50)(19,51)(20,54)(22,56)(23,25)(24,55)(29,49)(30,58)(32,60)(33,61)(35,57)(36,38)(37,62)(42,47)(43,59)(46,64)(48,52)(53,63)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 365273);