#6344278

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343678
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
prime radical	#62834	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ #3444 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2988 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,e^2=h,f^2=i,g^2=hi,b^a=bf,e^a=ei,f^a=fh,f^b=fi,g^b=ghi,d^c=di,e^c=eh

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, 6251474);