#6394050

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342836
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	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2883 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18,22,25,21,26,19,23)(20,28,27,32,29,24,30,31);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(17,19,21,22)(18,23,26,25)(20,27,29,30)(24,31,28,32);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20,19,27,21,29,22,30)(18,24,25,32,26,28,23,31);(3,12)(7,10)(11,13)(15,16);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fgh,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,g^c=gi,h^c=hi

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, 6301246);