#433427

order	512
nilpotency class	3
isoclinism	isoclinic class of #417581
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1372 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 340623);