#6497576

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	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1588 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)(17,18,23,27)(19,29,32,40)(20,33,37,44)(21,26,22,28)(24,38,41,31)(25,35,43,36)(30,42,34,39)(45,48,47,46);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,19)(18,24)(20,34)(21,38)(22,31)(23,32)(25,42)(26,29)(27,41)(28,40)(30,37)(33,47)(35,46)(36,48)(39,43)(44,45);(3,9)(6,13)(10,15)(14,16)(17,20,22,36,23,37,21,35)(18,25,26,33,27,43,28,44)(19,30,31,46,32,34,38,48)(24,39,29,45,41,42,40,47);(3,9)(6,13)(10,15)(14,16)(17,21,23,22)(18,26,27,28)(19,31,32,38)(20,35,37,36)(24,40,41,29)(25,33,43,44)(30,46,34,48)(39,47,42,45);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(18,27)(24,41)(25,43)(26,28)(29,40)(33,44)(39,42)(45,47)

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=i,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, 6404772);