#9028244

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

4899 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=ghi,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8935440);