#6526814

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500474
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	C₂²×C₄	C₂⁵
prime radical	#60559	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1661 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6434010);