#9140582

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

5098 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=ghi,d^2=gh,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, 9047778);