#463395

order	512
nilpotency class	3
isoclinism	isoclinic class of #462392
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2517
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16470
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

754 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 370591);