#9514942

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

4673 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,e^2=i,f^2=g,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, 9422138);