#9232823

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

4717 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)(9,10)(11,13)(12,16)(14,15)(17,18,23,27)(19,24,29,22)(20,25,30,31)(21,26,28,32);(3,7)(5,8)(10,15)(11,16)(17,19)(18,24)(20,21)(22,27)(23,29)(25,26)(28,30)(31,32);(1,2)(3,5)(4,6)(7,8)(9,11,14,16)(10,12,15,13)(17,20)(18,25)(19,28)(21,29)(22,26)(23,30)(24,32)(27,31);(9,12,14,13)(10,11,15,16)(17,21,23,28)(18,26,27,32)(19,20,29,30)(22,31,24,25);(1,3,4,7)(2,5,6,8)(9,10)(11,12)(13,16)(14,15)(17,20)(18,25)(19,21)(22,32)(23,30)(24,26)(27,31)(28,29);(1,2)(3,5)(4,6)(7,8)(9,13,14,12)(10,16,15,11)(17,22,23,24)(18,19,27,29)(20,26,30,32)(21,25,28,31)

Polycyclic presentation:

2^9|a^2=hi,c^2=g,d^2=i,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,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, 9140019);