#7563323

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

2690 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)(7,21)(8,16)(9,17)(10,30)(11,19)(12,18)(13,20)(14,24)(15,28)(22,31)(23,32)(25,35)(26,34)(27,33)(29,36);(1,2)(3,4)(5,7,11,24,13,26,12,25)(6,14,18,21,20,35,19,34)(8,27,28,22,10,23,9,29)(15,32,16,31,17,33,30,36);(5,8,12,9,13,10,11,28)(6,15,19,30,20,17,18,16)(7,22,24,23,26,29,25,27)(14,31,21,33,35,36,34,32);(2,4)(5,9,11,8,13,28,12,10)(6,16,18,17,20,30,19,15)(7,23,25,22,26,27,24,29)(14,32,34,36,35,33,21,31);(5,10,12,28,13,8,11,9)(6,17,19,16,20,15,18,30)(7,23,25,22,26,27,24,29)(14,33,34,31,35,32,21,36)

Polycyclic presentation:

2^9|b^2=f,c^2=gi,d^2=g,e^2=fi,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, 7470519);