#9127455

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

5044 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9034651);