#6669915

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	#61182	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1672 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)(19,25)(20,35)(21,27)(22,28)(23,29)(24,30)(26,38)(31,48)(32,42)(33,43)(34,45)(36,41)(37,46)(39,47)(40,44);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19)(18,25)(20,36)(21,32)(22,38)(23,40)(24,34)(26,28)(27,42)(29,44)(30,45)(31,37)(33,39)(35,41)(43,47)(46,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20,24,33)(18,26,30,46)(19,31,34,22)(21,37,23,38)(25,41,45,47)(27,35,29,43)(28,42,48,44)(32,39,40,36);(2,8)(6,9)(7,14)(13,15)(17,21,24,23)(18,27,30,29)(19,32,34,40)(20,37,33,38)(22,36,31,39)(25,42,45,44)(26,35,46,43)(28,41,48,47);(3,11)(6,9)(10,16)(13,15)(17,22,24,31)(18,28,30,48)(19,33,34,20)(21,39,23,36)(25,43,45,35)(26,42,46,44)(27,47,29,41)(32,37,40,38)

Polycyclic presentation:

2^9|b^2=f,c^2=fi,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, 6577111);