#6444326

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

2659 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)(5,6,11,12,10,16,9,15)(7,17,23,27,18,13,20,25)(8,14,24,26,21,30,22,29)(19,31,34,36,32,28,33,35);(6,12)(7,18)(8,21)(9,11)(13,27)(14,29)(15,16)(17,25)(19,32)(26,30)(28,36)(31,35);(1,2,3,4)(5,7,11,23,10,18,9,20)(6,13,12,25,16,17,15,27)(8,19,24,34,21,32,22,33)(14,28,26,35,30,31,29,36);(5,8,9,22,10,21,11,24)(6,14,15,29,16,30,12,26)(7,19,20,33,18,32,23,34)(13,28,27,36,17,31,25,35);(7,18)(13,17)(19,32)(20,23)(25,27)(28,31)(33,34)(35,36)

Polycyclic presentation:

2^9|a^2=fi,c^2=fg,d^2=f,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, 6351522);