#406110

order	512
nilpotency class	3
isoclinism	isoclinic class of #406110
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14924
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

978 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eh,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,f^c=fi,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, 313306);