#6669918

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	#3488	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1674 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)(19,30)(20,35)(21,29)(22,27)(23,28)(24,33)(25,38)(26,40)(31,43)(32,45)(34,46)(36,42)(37,47)(39,48)(41,44);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19)(18,24)(20,36)(21,31)(22,38)(23,33)(25,44)(26,43)(27,47)(28,30)(29,46)(32,39)(34,40)(35,48)(37,41)(42,45);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20,23,32)(18,25,28,47)(19,22,33,41)(21,37,40,38)(24,42,30,48)(26,45,29,35)(27,46,44,43)(31,36,34,39);(2,8)(6,9)(7,14)(13,15)(17,21,23,40)(18,26,28,29)(19,31,33,34)(20,37,32,38)(22,36,41,39)(24,43,30,46)(25,45,47,35)(27,48,44,42);(3,11)(6,9)(10,16)(13,15)(17,22)(18,27)(19,32)(20,33)(21,39)(23,41)(24,35)(25,46)(26,42)(28,44)(29,48)(30,45)(31,37)(34,38)(36,40)(43,47)

Polycyclic presentation:

2^9|b^2=fi,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,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, 6577114);