#8505266

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4752 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=i,d^2=hi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8412462);