#6342457

order	512
nilpotency class	2
isoclinism	isoclinic class of #3484
automorphism	Aut(G) order 131072
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₄²	C₂⁵
commutator, socle	C₂²	C₂³×C₄²
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1992 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)(3,6)(4,8)(5,7)(9,13)(10,12)(11,14)(15,16)(17,18,21,26,23,25,22,19)(20,28,30,27,29,24,31,32);(1,3,4,9,5,10,11,15)(2,6,8,13,7,12,14,16)(17,19,22,25,23,26,21,18)(20,27,31,28,29,32,30,24);(2,7)(6,12)(8,14)(13,16)(17,20)(18,24)(19,27)(21,30)(22,31)(23,29)(25,28)(26,32);(20,29)(24,28)(27,32)(30,31);(18,25)(19,26)(24,28)(27,32)

Polycyclic presentation:

2^9|a^2=f,b^2=g,f^2=h,g^2=i,c^a=ci,e^a=eh,e^b=eh,d^c=dh

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6249653);