#6444406

order	512
nilpotency class	3
isoclinism	isoclinic class of #6444406
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

2692 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:

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

Polycyclic presentation:

2^9|a^2=f,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^c=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, 6351602);