#6677484

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667760
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#3013	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1633 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=f,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi,h^c=hi,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6584680);