#6343506

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343418
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	C₂²×C₄	C₂⁵
prime radical	#62781	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2933 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 6250702);