#9032009

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

5285 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=i,e^2=g,f^2=gi,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, 8939205);