#448775

order	512
nilpotency class	3
isoclinism	isoclinic class of #448775
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15919
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

713 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:

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

Polycyclic presentation:

2^9|b^2=ef,c^2=e,d^2=eg,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,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, 355971);