#7563296

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519063
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2629 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=gi,d^2=g,e^2=fi,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,h^b=hi,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7470492);