#6444409

order	512
nilpotency class	3
isoclinism	isoclinic class of #6444408
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

2642 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:

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

Polycyclic presentation:

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

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, 6351605);