#7519081

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519075
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 ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2516 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=i,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,e^b=ei,h^b=hi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7426277);