#6443320

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437935
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4907 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)(7,14,19,27)(8,13,21,30)(9,23,10,15)(12,24,28,18)(17,29,33,36)(20,35,34,26)(22,32,25,31);(5,7,10,24,11,19,9,18)(6,12,23,27,16,28,15,14)(8,20,25,33,21,34,22,17)(13,29,32,26,30,36,31,35);(1,2,3,4)(5,8)(6,13)(7,17)(9,22)(10,25)(11,21)(12,26)(14,29)(15,31)(16,30)(18,34)(19,33)(20,24)(23,32)(27,36)(28,35);(5,7)(6,14)(8,17)(9,18)(10,24)(11,19)(12,23)(13,29)(15,28)(16,27)(20,25)(21,33)(22,34)(26,32)(30,36)(31,35);(8,21)(13,30)(17,33)(20,34)(22,25)(26,35)(29,36)(31,32)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,h^b=hi,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, 6350516);