#6347570

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347544
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	#62823	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2702 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)(4,5)(8,15)(9,10)(11,12)(14,16)(17,18,23,28)(19,29,33,38)(20,25,34,40)(21,27,22,26)(24,31,39,32)(30,43,44,48)(35,42,36,41)(37,45,47,46);(1,2,6,10)(3,8,13,16)(4,7,5,9)(11,15,12,14)(17,19,23,33)(18,24,28,39)(20,30,34,44)(21,32,22,31)(25,37,40,47)(26,29,27,38)(35,46,36,45)(41,43,42,48);(1,3)(2,8)(4,12)(5,11)(6,13)(7,14)(9,15)(10,16)(17,20)(18,25)(19,30)(21,35)(22,36)(23,34)(24,37)(26,41)(27,42)(28,40)(29,43)(31,45)(32,46)(33,44)(38,48)(39,47);(1,4,6,5)(2,7,10,9)(3,11,13,12)(8,15,16,14)(20,34)(25,40)(30,44)(35,36)(37,47)(41,42)(43,48)(45,46);(1,4,6,5)(2,9,10,7)(3,12,13,11)(8,15,16,14)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,34,36)(24,38,39,29)(25,41,40,42)(30,45,44,46)(37,48,47,43)

Polycyclic presentation:

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

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, 6254766);