#6343884

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343485
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	#62810	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2851 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6251080);