#6444307

order	512
nilpotency class	3
isoclinism	isoclinic class of #6439035
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

2685 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,h^b=hi,e^c=ei,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, 6351503);