#6343217

order	512
nilpotency class	3
isoclinism	isoclinic class of #61629
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂³×C₂²⋊C₄
prime radical	#62752	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #3444 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2737 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6250413);