#8720785

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

5031 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)(3,8)(4,5)(6,7)(9,10,13,15)(11,14,16,12)(17,18)(19,21)(20,28)(22,25)(23,27)(24,29)(26,32)(30,31);(1,3,6,5)(2,4,7,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,32)(28,30)(29,31);(1,3,6,5)(2,4,7,8)(9,10)(11,12)(13,15)(14,16)(20,29)(24,28)(26,31)(30,32);(2,7)(3,5)(9,11,13,16)(10,14,15,12)(17,20)(18,24)(19,26)(21,30)(22,29)(23,32)(25,28)(27,31);(1,4,6,8)(2,3,7,5)(9,10)(11,14)(12,16)(13,15)(17,21,22,23)(18,19,25,27)(20,30,29,32)(24,26,28,31);(1,5,6,3)(2,4,7,8)(9,12,13,14)(10,11,15,16)(17,19)(18,21)(20,31)(22,27)(23,25)(24,32)(26,29)(28,30)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=hi,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8627981);