#6498487

order	512
nilpotency class	3
isoclinism	isoclinic class of #6473153
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	#60799	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1728 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|c^2=gh,e^2=hi,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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6405683);