#9490528

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

4942 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=gh,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, 9397724);