#6444323

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438880
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

2646 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)(5,6,12,16,11,18,10,17)(7,19,22,13,24,36,23,32)(8,25,29,34,27,14,9,15)(20,35,21,33,26,31,28,30);(5,7)(6,13)(8,26)(9,28)(10,22)(11,24)(12,23)(14,33)(15,35)(16,19)(17,36)(18,32)(20,27)(21,29)(25,30)(31,34);(1,2,3,4)(5,8,12,29,11,27,10,9)(6,14,16,15,18,25,17,34)(7,20,22,21,24,26,23,28)(13,30,36,35,32,33,19,31);(5,9,10,27,11,29,12,8)(6,15,17,14,18,34,16,25)(7,21,23,20,24,28,22,26)(13,31,19,33,32,35,36,30);(5,10,11,12)(6,16,18,17)(7,22,24,23)(8,9,27,29)(13,19,32,36)(14,15,25,34)(20,21,26,28)(30,31,33,35)

Polycyclic presentation:

2^9|a^2=fi,c^2=fg,d^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,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, 6351519);