#7519074

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 ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2566 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,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,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, 7426270);