#6435640

order	512
nilpotency class	3
isoclinism	isoclinic class of #29592
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1770 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)(4,5)(7,8)(11,15)(14,16)(17,18,21,25,22,26,20,24)(19,27,28,31,29,32,30,23);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,27)(25,32)(26,31);(9,11,12,15)(10,14,13,16);(17,20,22,21)(18,24,26,25)(19,28,29,30)(23,27,31,32)

Polycyclic presentation:

2^9|a^2=g,c^2=f,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,c^b=ch,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6342836);