#6669714

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	#3488	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,28)(20,34)(21,27)(22,26)(23,32)(24,39)(25,40)(29,46)(30,31)(33,44)(35,41)(36,37)(38,43)(42,47)(45,48);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19)(18,23)(20,35)(21,30)(22,32)(24,42)(25,31)(26,28)(27,44)(29,39)(33,40)(34,45)(36,47)(37,46)(38,48)(41,43);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20,22,38)(18,24,26,37)(19,29,32,47)(21,36,40,39)(23,41,28,45)(25,43,27,34)(30,35,33,48)(31,46,44,42);(2,8)(6,9)(7,14)(13,15)(17,21,22,40)(18,25,26,27)(19,30,32,33)(20,36,38,39)(23,31,28,44)(24,43,37,34)(29,35,47,48)(41,46,45,42);(3,11)(6,9)(10,16)(13,15)(17,18)(19,31)(20,37)(21,25)(22,26)(23,33)(24,38)(27,40)(28,30)(29,42)(32,44)(34,36)(35,41)(39,43)(45,48)(46,47)

Polycyclic presentation:

2^9|b^2=fi,c^2=f,d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,e^c=ei,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, 6576910);