#6477426

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470058
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60752	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2937 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)(3,6)(4,7)(5,8)(9,13)(10,14)(11,15)(12,16)(17,18)(19,29)(20,33)(21,26)(22,27)(23,28)(24,31)(25,34)(30,38)(32,39)(35,41)(36,40)(37,42)(43,45)(44,47)(46,48);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,19,23,32)(18,24,28,40)(20,30,35,46)(21,36,22,31)(25,38,42,48)(26,29,27,39)(33,47,41,43)(34,45,37,44);(1,4,5,12)(2,7,8,16)(3,9,10,11)(6,13,14,15)(17,20)(18,25)(19,30)(21,34)(22,37)(23,35)(24,38)(26,41)(27,33)(28,42)(29,43)(31,44)(32,46)(36,45)(39,47)(40,48);(3,10)(6,14)(9,11)(13,15)(17,21,23,22)(18,26,28,27)(19,31,32,36)(20,34,35,37)(24,39,40,29)(25,41,42,33)(30,44,46,45)(38,47,48,43);(2,8)(6,14)(7,16)(13,15)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,f^b=fi,g^c=gi

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, 6384622);