#6343541

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	#62831	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2766 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,e^2=i,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, 6250737);