#6497575

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

1594 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)(17,18,23,26)(19,29,31,44)(20,33,37,43)(21,28,22,27)(24,38,40,32)(25,35,42,36)(30,41,34,39)(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)(22,32)(23,31)(25,41)(26,40)(27,44)(28,29)(30,37)(33,47)(35,46)(36,48)(39,42)(43,45);(3,9)(6,13)(10,15)(14,16)(17,20,22,36,23,37,21,35)(18,25,28,33,26,42,27,43)(19,30,32,46,31,34,38,48)(24,39,29,45,40,41,44,47);(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)(2,7,8,11)(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|a^2=h,c^2=g,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, 6404771);