#6669708

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	#61171	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1688 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)(20,33)(21,39)(22,27)(23,26)(24,32)(25,38)(29,43)(30,42)(31,40)(34,45)(35,36)(37,44)(41,46)(47,48);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,22,32)(18,24,27,28)(20,34,37,46)(21,30,23,40)(25,43,36,48)(26,42,39,31)(29,38,47,35)(33,41,44,45);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20,22,37)(18,25,27,36)(19,29,32,47)(21,35,23,38)(24,41,28,45)(26,44,39,33)(30,46,40,34)(31,48,42,43);(2,8)(6,9)(7,14)(13,15)(17,21,22,23)(18,26,27,39)(19,30,32,40)(20,35,37,38)(24,42,28,31)(25,44,36,33)(29,46,47,34)(41,43,45,48);(3,11)(6,9)(10,16)(13,15)(17,18)(19,31)(20,36)(21,26)(22,27)(23,39)(24,30)(25,37)(28,40)(29,43)(32,42)(33,35)(34,41)(38,44)(45,46)(47,48)

Polycyclic presentation:

2^9|b^2=f,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, 6576904);