#476966

order	512
nilpotency class	3
isoclinism	isoclinic class of #476082
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1214 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^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,d^c=de,f^c=fi,g^c=gi,h^c=hi,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, 384162);