#6444329

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438886
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 ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2672 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)(5,6,11,17,12,18,10,16)(7,19,22,31,23,32,28,13)(8,14,25,33,26,30,21,36)(9,15,24,35,27,29,20,34);(5,7)(6,13)(8,24)(9,25)(10,22)(11,28)(12,23)(14,15)(16,19)(17,32)(18,31)(20,26)(21,27)(29,30)(33,34)(35,36);(1,2,3,4)(5,8,11,25,12,26,10,21)(6,14,17,33,18,30,16,36)(7,20,22,9,23,24,28,27)(13,29,19,34,31,15,32,35);(5,9,11,24,12,27,10,20)(6,15,17,35,18,29,16,34)(7,21,22,8,23,25,28,26)(13,30,19,36,31,14,32,33);(5,10,12,11)(6,16,18,17)(7,22,23,28)(8,25,26,21)(9,20,27,24)(13,19,31,32)(14,33,30,36)(15,34,29,35)

Polycyclic presentation:

2^9|a^2=f,c^2=fg,d^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^c=ei,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, 6351525);