#6432119

order	512
nilpotency class	3
isoclinism	isoclinic class of #23439
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#121
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3114 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339315);