#498439

order	512
nilpotency class	3
isoclinism	isoclinic class of #365053
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2363
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#23084
prime radical	#3050	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

516 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:

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

Polycyclic presentation:

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

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, 405635);