#476280

order	512
nilpotency class	3
isoclinism	isoclinic class of #476082
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1606 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383476);