#436183

order	512
nilpotency class	3
isoclinism	isoclinic class of #419463
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2557
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15550
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

918 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)(4,20,24,41)(5,11)(6,13)(7,14)(8,12)(10,23,35,29)(16,33)(17,31,47,36)(18,40,28,32)(21,42)(22,48,51,62)(25,38)(26,39)(27,37)(30,44)(34,50,59,53)(43,58)(45,55)(46,56,52,61)(49,54)(57,64)(60,63);(1,3)(2,9)(4,21)(5,17)(6,18)(7,28)(8,19)(10,33)(11,36)(12,15)(13,32)(14,40)(16,23)(20,30)(22,49)(24,44)(25,46)(26,52)(27,47)(29,45)(31,37)(34,58)(35,55)(38,61)(39,56)(41,42)(43,50)(48,60)(51,63)(53,64)(54,62)(57,59);(1,4)(2,10,12,35)(3,16)(5,22)(6,23)(7,29)(8,24)(9,30,15,42)(11,34,37,59)(13,20,14,41)(17,43)(18,44)(19,45)(21,28)(25,50)(26,53)(27,51)(31,54,36,60)(32,55,40,33)(38,48,39,62)(46,63)(47,64)(49,52)(56,58,61,57);(1,5,6,25,8,27,7,26)(2,11,13,38,12,37,14,39)(3,17,18,46,19,47,28,52)(4,22,23,50,24,51,29,53)(9,31,32,56,15,36,40,61)(10,34,20,48,35,59,41,62)(16,43,44,63,45,64,21,49)(30,54,55,58,42,60,33,57)

Polycyclic presentation:

2^9|a^2=fg,c^2=e,d^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,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, 343379);