#7563346

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563346
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

2681 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)(7,19)(8,25)(9,14)(10,17)(11,16)(12,18)(13,22)(15,28)(20,29)(21,30)(23,32)(24,31)(26,35)(27,34)(33,36);(1,2)(3,4)(5,7,10,22,12,24,11,23)(6,13,16,19,18,32,17,31)(8,26,9,20,28,36,27,21)(14,33,25,29,34,35,15,30);(5,8,10,9,12,28,11,27)(6,14,16,25,18,34,17,15)(7,20,23,26,24,21,22,36)(13,29,31,33,32,30,19,35);(2,4)(5,8,10,9,12,28,11,27)(6,14,16,25,18,34,17,15)(7,21,23,36,24,20,22,26)(13,30,31,35,32,29,19,33);(5,9,10,28,12,27,11,8)(6,15,16,14,18,25,17,34)(7,20,22,36,24,21,23,26)(13,30,19,33,32,29,31,35)

Polycyclic presentation:

2^9|b^2=f,c^2=g,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,e^c=ei,e^d=ei

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, 7470542);