#6358964

order	512
nilpotency class	3
isoclinism	isoclinic class of #6358842
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	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1570 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,d^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=eh,f^b=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6266160);