#8599368

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

5070 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8506564);