#9161768

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

5015 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=gh,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9068964);