#451923

order	512
nilpotency class	3
isoclinism	isoclinic class of #420055
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2476
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16021
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

746 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)(3,14)(4,19,24,47)(5,11,28,40)(6,29)(7,12)(8,13)(9,17)(10,23,37,22)(15,31,20,35)(16,44,45,38)(18,33)(21,46)(25,54,56,32)(26,57,27,39)(30,34)(36,51)(41,43,59,49)(42,60)(48,58)(50,61)(52,62)(53,64)(55,63);(1,3)(2,9)(4,20)(5,25)(6,30)(7,17)(8,18)(10,35)(11,38)(12,33)(13,34)(14,29)(15,24)(16,26)(19,41)(21,50)(22,43)(23,49)(27,45)(28,56)(31,37)(32,39)(36,61)(40,44)(42,52)(46,60)(47,59)(48,51)(53,55)(54,57)(58,64)(62,63);(1,4,6,22,8,24,7,23)(2,10,12,19,13,37,29,47)(3,15,17,43,18,20,30,49)(5,21,26,51,28,53,27,52)(9,31,14,41,34,35,33,59)(11,36,39,46,40,62,57,64)(16,42,25,55,45,48,56,50)(32,58,38,63,54,60,44,61);(1,5)(2,11,13,40)(3,16,18,45)(4,21)(6,26)(7,27)(8,28)(9,32)(10,36,37,62)(12,39,29,57)(14,38)(15,42,20,48)(17,25,30,56)(19,46,47,64)(22,51)(23,52)(24,53)(31,58)(33,44)(34,54)(35,60)(41,63)(43,55,49,50)(59,61)

Polycyclic presentation:

2^9|a^2=efgh,c^2=e,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi,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, 359119);