#6349415

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349370
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	#62840	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2628 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:

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

Polycyclic presentation:

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

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, 6256611);