#471017

order	512
nilpotency class	3
isoclinism	isoclinic class of #471017
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16731
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

648 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)(3,12)(4,9,16,24)(5,8)(6,11)(7,14)(10,18)(13,20,27,25)(15,21)(17,22,29,23)(19,28)(26,30,32,31);(1,3)(2,7,11,12)(4,13)(5,15)(6,14)(8,19,10,21)(9,20,24,25)(16,27)(17,26)(18,28)(22,30,23,31)(29,32);(2,8)(4,16)(5,18)(7,19)(9,23)(10,11)(12,21)(15,28)(20,30)(22,24)(25,31)(26,32);(1,4,5,17,6,16,18,29)(2,9,10,23,11,24,8,22)(3,13,15,26,14,27,28,32)(7,20,21,31,12,25,19,30)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=f,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,e^b=ei,h^b=hi,d^c=de,f^c=fi,e^d=ei,g^d=gi,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, 378213);