#7614091

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

2504 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)(8,15)(9,27)(10,17)(11,13)(12,22)(14,26)(16,28)(19,34)(20,30)(21,32)(23,31)(24,29)(25,33)(35,36);(2,3)(5,7,10,22,11,19,28,23)(6,12,16,18,13,31,17,34)(8,24,25,20,26,35,9,21)(14,32,33,36,15,30,27,29);(1,2,4,3)(6,13)(7,19)(14,15)(16,17)(20,21)(22,23)(24,35)(27,33);(1,3,4,2)(5,8,10,25,11,26,28,9)(6,14,17,27,13,15,16,33)(7,20,23,24,19,21,22,35)(12,29,18,32,31,36,34,30);(5,9,28,26,11,25,10,8)(6,15,17,33,13,14,16,27)(7,21,23,35,19,20,22,24)(12,30,34,36,31,32,18,29)

Polycyclic presentation:

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

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, 7521287);