#436732

order	512
nilpotency class	3
isoclinism	isoclinic class of #418818
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2684
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15568
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

817 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,d^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,g^d=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, 343928);