#6435856

order	512
nilpotency class	3
isoclinism	isoclinic class of #29742
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1772 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6343052);