#8639538

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
prime radical	#2430	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4244 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(9,10)(11,12)(13,15)(14,16)(17,20)(18,19)(21,23)(22,24)(25,26,31,33)(27,29,30,32)(28,36,39,34)(35,40,37,38);(2,5)(6,8)(9,11,15,17)(10,12,13,20)(14,18,23,24)(16,19,21,22)(25,27,31,30)(26,32,33,29)(28,37,39,35)(34,38,36,40);(1,2,4,5)(3,6,7,8)(10,13)(11,17)(16,21)(18,24)(26,33)(29,32)(34,36)(38,40);(3,7)(6,8)(9,12)(10,11)(13,17)(14,22)(15,20)(16,24)(18,21)(19,23)(25,28,31,39)(26,34,33,36)(27,35,30,37)(29,38,32,40);(1,2)(3,6)(4,5)(7,8)(9,13,15,10)(11,12,17,20)(14,21,23,16)(18,19,24,22)(25,29,31,32)(26,27,33,30)(28,38,39,40)(34,35,36,37);(1,3,4,7)(2,6,5,8)(9,14)(10,16)(11,18)(12,19)(13,21)(15,23)(17,24)(20,22)(25,30,31,27)(26,32,33,29)(28,35,39,37)(34,38,36,40)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^2=g,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8546734);