#9032029

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

4950 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)(17,18,23,27)(19,21,28,24)(20,25,30,32)(22,26,31,29);(3,7)(5,8)(9,10,13,12)(11,14,15,16)(17,19)(18,24)(20,22)(21,27)(23,28)(25,29)(26,32)(30,31);(1,2)(3,5)(4,6)(7,8)(10,12)(11,15)(17,20,23,30)(18,25,27,32)(19,22,28,31)(21,26,24,29);(9,11,13,15)(10,14,12,16)(17,21,23,24)(18,19,27,28)(20,29,30,26)(22,25,31,32);(1,3)(2,5)(4,7)(6,8)(9,11)(10,14)(12,16)(13,15)(17,20)(18,25)(19,22)(21,26)(23,30)(24,29)(27,32)(28,31);(1,2)(3,5)(4,6)(7,8)(9,12,13,10)(11,16,15,14)(17,22)(18,26)(19,20)(21,25)(23,31)(24,32)(27,29)(28,30)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8939225);