#436354

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

924 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)(3,15,19,9)(4,20,25,41)(5,26)(6,13)(7,14)(8,12)(10,23,37,24)(11,28)(16,35)(17,46,48,55)(18,34,30,33)(21,42)(22,51,54,36)(27,39)(29,40)(31,44)(32,47,38,56)(43,61)(45,58)(49,52,62,53)(50,60)(57,64)(59,63);(1,3)(2,9)(4,21)(5,17)(6,18)(7,30)(8,19)(10,35)(11,38)(12,15)(13,33)(14,34)(16,23)(20,31)(22,50)(24,45)(25,44)(26,55)(27,47)(28,56)(29,48)(32,39)(36,60)(37,58)(40,46)(41,42)(43,52)(49,61)(51,59)(53,64)(54,63)(57,62);(1,4)(2,10,12,37)(3,16)(5,22)(6,23)(7,24)(8,25)(9,31,15,42)(11,36,39,51)(13,20,14,41)(17,43)(18,44)(19,45)(21,30)(26,49,40,62)(27,52)(28,53)(29,54)(32,57,38,61)(33,58,34,35)(46,60,55,59)(47,63)(48,64)(50,56);(1,5,6,27,8,29,7,28)(2,11,14,40,12,39,13,26)(3,17,18,47,19,48,30,56)(4,22,23,52,25,54,24,53)(9,32,34,55,15,38,33,46)(10,36,41,62,37,51,20,49)(16,43,44,63,45,64,21,50)(31,57,35,59,42,61,58,60)

Polycyclic presentation:

2^9|a^2=fg,c^2=e,d^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,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, 343550);