#6444411

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

2726 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)(7,17,20,12,19,28,21,27)(8,13,24,30,23,29,25,22)(18,31,33,26,35,36,34,32);(1,2)(3,4)(5,6)(7,17)(8,22)(9,15)(10,14)(11,16)(12,21)(13,25)(18,32)(19,28)(20,27)(23,30)(24,29)(26,35)(31,34)(33,36);(2,4)(5,7)(6,12,16,27)(8,18)(9,20)(10,21)(11,19)(13,26,29,32)(14,28,15,17)(22,31,30,36)(23,35)(24,33)(25,34);(2,4)(5,8,9,24,11,23,10,25)(6,13,14,30,16,29,15,22)(7,18,20,33,19,35,21,34)(12,26,28,36,27,32,17,31);(7,19)(8,23)(12,27)(13,29)(17,28)(20,21)(22,30)(24,25)

Polycyclic presentation:

2^9|a^2=f,c^2=fgi,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,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, 6351607);