#482039

order	512
nilpotency class	4
isoclinism	isoclinic class of #482033
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1519
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂×M₁₆	C₂⁴
prime radical	#60461	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1006 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389235);