#6438906

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438903
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2424
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3074 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6346102);