#6686101

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682628
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60706	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1638 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,9)(4,8)(5,7)(6,11)(10,16)(12,14)(13,15)(17,18,23,27)(19,29,31,24)(20,33,32,44)(21,28,34,25)(22,26,38,43)(30,46,47,40)(35,42,37,48)(36,45,39,41);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19,23,31)(18,24,27,29)(20,34,32,21)(22,37,38,35)(25,33,28,44)(26,42,43,48)(30,39,47,36)(40,41,46,45);(17,20,34,19,23,32,21,31)(18,24,28,44,27,29,25,33)(22,35,39,30,38,37,36,47)(26,40,45,48,43,46,41,42);(2,7)(6,9)(8,15)(14,16)(17,21,23,34)(18,25,27,28)(19,20,31,32)(22,36,38,39)(24,33,29,44)(26,41,43,45)(30,35,47,37)(40,42,46,48);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,22,23,38)(18,26,27,43)(19,30,31,47)(20,35,32,37)(21,36,34,39)(24,40,29,46)(25,41,28,45)(33,42,44,48)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,d^2=i,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6593297);