#6669919

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667964
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	#3489	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1653 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)(19,24)(20,34)(21,26)(22,27)(23,29)(25,37)(28,39)(30,45)(31,42)(32,43)(33,44)(35,48)(36,47)(38,46)(40,41);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,23,33)(18,24,29,44)(20,35,38,40)(21,31,39,32)(22,37,30,36)(25,45,47,27)(26,42,28,43)(34,48,46,41);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20)(18,25)(19,30)(21,36)(22,33)(23,38)(24,41)(26,34)(27,43)(28,46)(29,47)(31,35)(32,40)(37,39)(42,45)(44,48);(2,8)(6,9)(7,14)(13,15)(17,21,23,39)(18,26,29,28)(19,31,33,32)(20,36,38,37)(22,40,30,35)(24,42,44,43)(25,34,47,46)(27,41,45,48);(3,11)(6,9)(10,16)(13,15)(17,22,23,30)(18,27,29,45)(19,20,33,38)(21,35,39,40)(24,34,44,46)(25,43,47,42)(26,48,28,41)(31,37,32,36)

Polycyclic presentation:

2^9|b^2=fi,c^2=f,d^2=i,e^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,g^c=gi,h^c=hi,e^d=ei,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, 6577115);