#6673595

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	#3494	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1645 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)(4,8)(5,7)(6,11)(10,16)(12,14)(13,15)(17,18)(19,30)(20,35)(21,28)(22,27)(23,40)(24,29)(25,33)(26,37)(31,45)(32,48)(34,43)(36,47)(38,46)(39,42)(41,44);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19,22,33)(18,25,27,30)(20,36,32,39)(21,38,31,37)(23,34,24,41)(26,45,46,28)(29,43,40,44)(35,42,48,47);(17,20,22,32)(18,26,27,46)(19,31,33,21)(23,38,24,37)(25,42,30,47)(28,43,45,44)(29,48,40,35)(34,39,41,36);(2,7)(6,9)(8,15)(14,16)(18,27)(25,30)(26,46)(28,45)(29,40)(35,48)(42,47)(43,44);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,21)(18,28)(19,32)(20,33)(22,31)(23,39)(24,36)(25,35)(26,44)(27,45)(29,47)(30,48)(34,38)(37,41)(40,42)(43,46)

Polycyclic presentation:

2^9|b^2=f,c^2=i,e^2=fi,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, 6580791);