#9101908

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

5526 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,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, 9009104);