#9137494

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

4723 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^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, 9044690);