#436355

order	512
nilpotency class	3
isoclinism	isoclinic class of #420517
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15556
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

912 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:

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

Polycyclic presentation:

2^9|a^2=fg,c^2=e,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,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, 343551);