#436027

order	512
nilpotency class	3
isoclinism	isoclinic class of #436027
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15546
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

780 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 343223);