#9720439

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

4675 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,e^2=gi,f^2=ghi,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, 9627635);