#6526817

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500482
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	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1639 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)(17,18,23,28)(19,29,32,42)(20,33,34,41)(21,27,22,26)(24,37,39,31)(25,35,40,36)(30,44,45,38)(43,47,48,46);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19)(18,24)(20,30)(21,37)(22,31)(23,32)(25,38)(26,42)(27,29)(28,39)(33,43)(34,45)(35,47)(36,46)(40,44)(41,48);(3,9)(4,11)(6,12)(7,14)(17,20,22,36,23,34,21,35)(18,25,27,33,28,40,26,41)(19,30,31,46,32,45,37,47)(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)(20,34)(25,40)(30,45)(33,41)(35,36)(38,44)(43,48)(46,47)

Polycyclic presentation:

2^9|a^2=hi,c^2=g,d^2=h,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,e^c=ei

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, 6434013);