#6343545

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343524
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	#3444	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2721 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:

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

Polycyclic presentation:

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

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, 6250741);