#6673800

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

1629 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)(19,28)(20,25)(21,36)(22,26)(23,27)(24,30)(29,43)(31,39)(32,42)(33,35)(34,40)(37,41)(38,45)(44,48)(46,47);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,18)(19,28)(20,33)(21,26)(22,36)(23,27)(24,31)(25,35)(29,38)(30,39)(32,42)(34,41)(37,40)(43,45)(44,47)(46,48);(17,19,23,32)(18,24,27,39)(20,29,34,44)(21,31,22,30)(25,38,40,47)(26,42,36,28)(33,43,41,48)(35,46,37,45);(2,7)(6,9)(8,15)(14,16)(20,34)(25,40)(29,44)(33,41)(35,37)(38,47)(43,48)(45,46);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,20)(18,25)(19,29)(21,35)(22,37)(23,34)(24,38)(26,41)(27,40)(28,43)(30,45)(31,46)(32,44)(33,36)(39,47)(42,48)

Polycyclic presentation:

2^9|b^2=f,c^2=i,e^2=f,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, 6580996);