#6439028

order	512
nilpotency class	3
isoclinism	isoclinic class of #6439025
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

3131 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)(5,6,12,16)(7,10,15,11)(8,19,24,17)(9,14,28,25)(13,22,20,23)(18,26,30,27)(21,32,36,31)(29,34,33,35);(5,7,11,6,12,15,10,16)(8,20,23,17,24,13,22,19)(9,25,27,18,28,14,26,30)(21,31,35,29,36,32,34,33);(2,4)(5,8,11,23,12,24,10,22)(6,13,15,19,16,20,7,17)(9,21,27,35,28,36,26,34)(14,29,30,32,25,33,18,31);(5,9)(6,14)(7,18)(8,21)(10,26)(11,27)(12,28)(13,29)(15,30)(16,25)(17,31)(19,32)(20,33)(22,34)(23,35)(24,36);(5,10,12,11)(6,15,16,7)(8,22,24,23)(9,26,28,27)(13,19,20,17)(14,30,25,18)(21,34,36,35)(29,32,33,31)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=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, 6346224);