#9101811

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

5655 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)(9,10,13,15)(11,12,16,14)(18,22)(23,26)(24,28)(30,32);(3,7)(6,8)(9,11,13,16)(10,14,15,12)(17,18,21,22)(19,23,25,26)(20,24,29,28)(27,30,31,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,15)(14,16)(19,25)(23,26)(27,31)(30,32);(9,12)(10,11)(13,14)(15,16)(17,18)(19,26)(20,28)(21,22)(23,25)(24,29)(27,30)(31,32);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,16)(14,15)(17,19,21,25)(18,23,22,26)(20,27,29,31)(24,30,28,32);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,14)(15,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9009007);