#483240

order	512
nilpotency class	4
isoclinism	isoclinic class of #483235
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1551
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#3508	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,e^2=h,f^2=ghi,g^2=i,b^a=be,c^a=cg,e^a=eg,g^a=gi,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,d^c=di,e^c=ei

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, 390436);