#6355533

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355483
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	#60546	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1597 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,15)(12,16)(17,18,23,28)(19,24,30,38)(20,33,35,42)(21,27,22,26)(25,36,41,34)(29,43,44,48)(31,40,32,39)(37,46,47,45);(1,3,5,13,6,12,4,11)(2,7,8,10,9,15,14,16)(17,19)(18,24)(20,29)(21,31)(22,32)(23,30)(25,37)(26,39)(27,40)(28,38)(33,43)(34,45)(35,44)(36,46)(41,47)(42,48);(17,20,23,35)(18,25,28,41)(19,29,30,44)(21,34,22,36)(24,37,38,47)(26,33,27,42)(31,45,32,46)(39,43,40,48);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,10,15,16)(19,30)(24,38)(29,44)(31,32)(37,47)(39,40)(43,48)(45,46);(3,12)(7,15)(10,16)(11,13)(17,21,23,22)(18,26,28,27)(19,31,30,32)(20,34,35,36)(24,39,38,40)(25,33,41,42)(29,45,44,46)(37,43,47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=h,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fh,g^a=gi,d^b=di,e^b=eh,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, 6262729);