#448887

order	512
nilpotency class	3
isoclinism	isoclinic class of #448887
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15924
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

846 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:

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

Polycyclic presentation:

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

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, 356083);