#6673801

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	#61267	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1648 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)(20,34)(21,25)(22,27)(23,31)(24,37)(26,38)(29,46)(30,32)(33,44)(35,40)(36,43)(39,42)(41,47)(45,48);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19,21,31)(18,23,25,28)(20,35,36,48)(22,32,38,33)(24,41,42,46)(26,30,27,44)(29,39,47,37)(34,45,43,40);(17,20,21,36)(18,24,25,42)(19,29,31,47)(22,39,38,37)(23,40,28,45)(26,43,27,34)(30,46,44,41)(32,35,33,48);(2,7)(6,9)(8,15)(14,16)(18,25)(23,28)(24,42)(26,27)(30,44)(34,43)(40,45)(41,46);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,18)(19,30)(20,24)(21,25)(22,26)(23,33)(27,38)(28,32)(29,46)(31,44)(34,37)(35,45)(36,42)(39,43)(40,48)(41,47)

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,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, 6580997);