#9064621

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

4790 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=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, 8971817);