#465309

order	512
nilpotency class	3
isoclinism	isoclinic class of #465309
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16518
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

773 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=eg,c^2=ef,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^b=fi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,h^d=hi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 372505);