#495943

order	512
nilpotency class	4
isoclinism	isoclinic class of #2238
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#481
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₄×#481
prime radical	#3054	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

565 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=eh,d^2=g,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,f^b=fi,e^c=ei,f^c=fhi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403139);