#6497582

order	512
nilpotency class	3
isoclinism	isoclinic class of #6473165
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

1566 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)(3,6,9,13)(4,11,12,7)(10,16,15,14)(17,18)(19,29)(20,34)(21,26)(22,27)(23,28)(24,32)(25,35)(30,44)(31,39)(33,42)(36,40)(37,41)(38,46)(43,45)(47,48);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(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)(6,13)(10,15)(14,16)(17,20,22,35,23,36,21,37)(18,25,26,34,28,41,27,40)(19,30,32,45,33,46,31,47)(24,38,29,43,39,44,42,48);(3,9)(6,13)(10,15)(14,16)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,36,37)(24,29,39,42)(25,40,41,34)(30,45,46,47)(38,48,44,43);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(20,36)(25,41)(30,46)(34,40)(35,37)(38,44)(43,48)(45,47)

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,d^c=di,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, 6404778);