#9716131

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

4742 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=g,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, 9623327);