#6356706

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356633
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#61986	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1512 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6263902);