#502477

order	512
nilpotency class	4
isoclinism	isoclinic class of #502475
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#25384
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#28249	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

637 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=ef,d^2=ei,e^2=i,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,f^c=fh,g^c=ghi,h^c=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, 409673);