#6495078

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470355
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35329	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1856 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=fg,c^2=g,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=di

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, 6402274);