#8943312

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	#35733	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4303 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)(11,19)(12,15)(13,17)(14,18)(16,20)(21,23)(22,24)(25,26)(27,29)(28,37)(30,38)(31,33)(32,36)(34,35)(39,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,16)(12,19)(13,14)(15,20)(17,22)(18,21)(23,24)(25,27,31,36)(26,32,33,29)(28,34,38,40)(30,35,37,39);(1,2,4,6)(3,7,8,5)(13,23)(14,24)(17,21)(18,22)(26,33)(29,32)(30,37)(35,39);(3,8)(5,7)(9,12,15,10)(11,19,20,16)(13,17,23,21)(14,22,24,18)(25,28)(26,30)(27,34)(29,39)(31,38)(32,35)(33,37)(36,40);(1,3)(2,5)(4,8)(6,7)(9,13)(10,17)(11,14)(12,21)(15,23)(16,22)(18,19)(20,24)(25,29,31,32)(26,27,33,36)(28,35,38,39)(30,40,37,34);(1,2)(3,5)(4,6)(7,8)(9,14)(10,18)(11,13)(12,22)(15,24)(16,21)(17,19)(20,23)(25,30)(26,28)(27,35)(29,40)(31,37)(32,34)(33,38)(36,39)

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8850508);