#6438903

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

3133 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)(3,12)(4,16)(5,20)(6,10)(7,11)(8,15)(9,19)(13,29)(14,25)(17,23)(18,27)(21,24)(22,28)(26,30)(31,32);(1,3,5,15,7,14,6,21)(2,8,10,12,11,24,20,25)(4,17,18,13,19,31,22,30)(9,26,27,32,16,29,28,23);(1,4,5,18,7,19,6,22)(2,9,10,27,11,16,20,28)(3,13,15,31,14,30,21,17)(8,23,12,26,24,32,25,29);(1,2)(3,12)(4,9)(5,10)(6,20)(7,11)(8,21)(13,26)(14,25)(15,24)(16,19)(17,23)(18,27)(22,28)(29,30)(31,32);(3,14)(8,24)(12,25)(13,30)(15,21)(17,31)(23,32)(26,29)

Polycyclic presentation:

2^9|b^2=f,c^2=f,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.

Relationships

maximal subgroup of	#72213603, #72214082, #72215906, …
maximal normal subgroup of	#72213603, #72214082, #72215906, …
maximal quotient of	#72311305

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6346099);