#6444324

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438880
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2678 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=fgi,d^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^c=hi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351520);