#409366

order	512
nilpotency class	3
isoclinism	isoclinic class of #409366
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15018
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

830 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 316562);