#8899082

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	#37817	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4300 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8806278);