#6496548

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470409
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	#3006	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1669 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)(19,29,33,41)(20,34,37,43)(21,26,22,28)(24,31,40,32)(25,38,44,36)(30,42,35,39)(45,46,48,47);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,19,21,32,23,33,22,31)(18,24,28,29,27,40,26,41)(20,35,36,46,37,30,38,47)(25,42,34,48,44,39,43,45);(3,9)(6,13)(10,15)(14,16)(17,20)(18,25)(19,30)(21,36)(22,38)(23,37)(24,39)(26,43)(27,44)(28,34)(29,45)(31,46)(32,47)(33,35)(40,42)(41,48);(3,9)(6,13)(10,15)(14,16)(17,21,23,22)(18,26,27,28)(19,31,33,32)(20,36,37,38)(24,29,40,41)(25,43,44,34)(30,46,35,47)(39,45,42,48);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(18,27)(24,40)(25,44)(26,28)(29,41)(34,43)(39,42)(45,48)

Polycyclic presentation:

2^9|a^2=h,b^2=f,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,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6403744);