#7563320

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

2684 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=gi,d^2=g,e^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,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, 7470516);