#6443030

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437793
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
prime radical	#60415	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2511 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,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, 6350226);