#6573988

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573652
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2499 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

(2,4)(5,6)(7,18)(8,21)(9,16)(10,11)(12,29)(13,17)(14,15)(19,34)(20,23)(22,35)(24,31)(25,27)(26,28)(30,33)(32,36);(1,2,3,4)(5,7,11,15)(6,14,10,18)(8,17,23,29)(9,19,26,31)(12,20,13,21)(16,24,28,34)(22,33,36,25)(27,32,30,35);(5,8,11,23)(6,15,10,7)(9,24,26,34)(12,18,13,14)(16,32,28,35)(17,20,29,21)(19,25,31,33)(22,30,36,27);(5,9,11,26)(6,16,10,28)(7,19,15,31)(8,22,23,36)(12,27,13,30)(14,24,18,34)(17,33,29,25)(20,32,21,35);(5,10)(6,11)(7,8)(9,25)(12,29)(13,17)(14,20)(15,23)(16,27)(18,21)(19,34)(22,32)(24,31)(26,33)(28,30)(35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481184);