#6344130

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	C₄²	C₂⁵
prime radical	C₂×#3511	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2995 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=h,f^2=i,g^2=hi,b^a=bf,c^a=chi,f^a=fh,f^b=fi,g^b=ghi,e^d=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, 6251326);