#6351877

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351869
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	C₄²	C₂⁵
prime radical	#60801	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2647 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,e^2=h,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=ghi,d^b=dh,f^b=fh,g^b=gi,e^c=eh,f^c=fi,g^c=ghi

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, 6259073);