#436029

order	512
nilpotency class	3
isoclinism	isoclinic class of #427666
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2527
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15546
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

733 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)(3,15,20,9,19,40,18,33)(4,21,23,36,24,10,29,35)(5,25,30,39,28,38,27,11)(16,42,44,31,46,41,45,50)(17,47,49,37,48,32,26,57)(22,52,54,51,55,56,58,34)(43,59,53,60,63,64,62,61);(1,3)(2,9)(4,16)(5,26)(6,20)(7,19)(8,18)(10,31)(11,37)(12,40)(13,33)(14,15)(17,27)(21,50)(22,53)(23,45)(24,46)(25,47)(28,49)(29,44)(30,48)(32,38)(34,60)(35,42)(36,41)(39,57)(43,54)(51,61)(52,59)(55,62)(56,64)(58,63);(1,4,6,23,7,24,8,29)(2,10,12,35,13,21,14,36)(3,16,18,44,19,46,20,45)(5,22,27,54,28,55,30,58)(9,31,15,41,33,50,40,42)(11,34,38,52,39,51,25,56)(17,43,26,53,48,63,49,62)(32,59,37,60,47,64,57,61);(1,5,6,27,7,28,8,30)(2,11,14,25,13,39,12,38)(3,17,18,26,19,48,20,49)(4,22,23,54,24,55,29,58)(9,32,40,57,33,47,15,37)(10,34,36,56,21,51,35,52)(16,43,44,53,46,63,45,62)(31,59,42,61,50,64,41,60)

Polycyclic presentation:

2^9|a^2=eg,c^2=e,d^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,g^c=gi

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, 343225);