#9506915

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

5277 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9414111);