#498583

order	512
nilpotency class	4
isoclinism	isoclinic class of #498581
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#23188
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

634 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=fi,d^2=ei,e^2=h,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,d^b=de,e^b=eh,g^b=gh,h^b=hi,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, 405779);