#6364609

order	512
nilpotency class	3
isoclinism	isoclinic class of #6364250
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

1524 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=h,e^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=ehi,f^b=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6271805);