#9061589

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

5295 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)(3,6)(4,5)(7,8)(11,12)(14,15)(17,18,23,25)(19,20,29,24)(21,26,31,30)(22,32,28,27);(1,3)(2,6)(4,7)(5,8)(9,10,13,16)(11,14,12,15)(17,19,23,29)(18,24,25,20)(21,28,31,22)(26,32,30,27);(1,3,5,8)(2,7,4,6)(9,11,13,12)(10,14,16,15)(18,25)(20,24)(21,31)(22,28);(3,8)(6,7)(17,20,23,24)(18,19,25,29)(21,32,31,27)(22,30,28,26);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,26)(19,28)(20,27)(22,29)(23,31)(24,32)(25,30);(1,4,5,2)(3,7,8,6)(9,12,13,11)(10,15,16,14)(17,22,23,28)(18,27,25,32)(19,21,29,31)(20,30,24,26)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=gh,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8968785);