#8990895

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	#35811	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4269 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=i,d^2=i,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8898091);