#9726264

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

4948 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=ghi,d^2=h,e^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9633460);