#6343202

order	512
nilpotency class	3
isoclinism	isoclinic class of #61569
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	#62781	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2882 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=ch,e^a=eh,f^a=fh,e^b=ei,f^b=fh,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, 6250398);