#7614090

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563319
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

2546 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)(7,17)(8,22)(9,26)(10,27)(11,14)(12,16)(13,21)(15,24)(18,32)(19,30)(20,33)(23,36)(25,35)(28,31)(29,34);(2,3)(5,7,10,21,11,18,12,28)(6,13,16,17,14,31,27,32)(8,23,9,20,24,34,25,19)(15,30,26,29,22,33,35,36);(1,2,4,3)(6,14)(7,18)(15,22)(16,27)(19,20)(21,28)(23,34)(26,35);(1,3,4,2)(5,8,12,25,11,24,10,9)(6,15,16,26,14,22,27,35)(7,19,21,23,18,20,28,34)(13,29,32,30,31,36,17,33);(5,9,10,24,11,25,12,8)(6,15,16,26,14,22,27,35)(7,20,21,34,18,19,28,23)(13,30,17,29,31,33,32,36)

Polycyclic presentation:

2^9|b^2=f,c^2=gi,d^2=h,e^2=f,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

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, 7521286);