#6444319

order	512
nilpotency class	3
isoclinism	isoclinic class of #25140
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2677 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)(5,6,11,17,12,18,10,16)(7,19,22,32,23,36,29,13)(8,24,9,15,27,14,26,34)(20,35,28,30,25,31,21,33);(5,7)(6,13)(8,25)(9,28)(10,22)(11,29)(12,23)(14,33)(15,35)(16,19)(17,36)(18,32)(20,27)(21,26)(24,30)(31,34);(1,2,3,4)(5,8,10,26,12,27,11,9)(6,14,16,15,18,24,17,34)(7,20,29,21,23,25,22,28)(13,30,36,35,32,33,19,31);(5,9,11,27,12,26,10,8)(6,15,17,14,18,34,16,24)(7,21,22,20,23,28,29,25)(13,31,19,33,32,35,36,30);(5,10,12,11)(6,16,18,17)(7,22,23,29)(8,26,27,9)(13,19,32,36)(14,15,24,34)(20,28,25,21)(30,31,33,35)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351515);