#433965

order	512
nilpotency class	3
isoclinism	isoclinic class of #425083
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2576
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15494
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

973 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:

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

Polycyclic presentation:

2^9|a^2=eg,c^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=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, 341161);