#8759103

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	#49769	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4200 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)(11,16)(12,21)(13,14)(15,18)(17,24)(19,23)(20,22)(25,26,31,34)(27,35,38,29)(28,33,30,32)(36,39,37,40);(1,2)(3,7)(4,5)(6,8)(9,11,15,12)(10,16,18,21)(13,22,17,19)(14,20,24,23)(25,27,31,38)(26,29,34,35)(28,36,30,37)(32,39,33,40);(2,5)(6,7)(9,12,15,11)(10,16,18,21)(13,22,17,19)(14,23,24,20)(25,27)(26,29)(28,36)(30,37)(31,38)(32,39)(33,40)(34,35);(1,2)(3,6)(4,5)(7,8)(25,28,31,30)(26,32,34,33)(27,36,38,37)(29,39,35,40);(1,2)(3,6)(4,5)(7,8)(9,13,15,17)(10,14,18,24)(11,19,12,22)(16,23,21,20)(25,29)(26,27)(28,39)(30,40)(31,35)(32,36)(33,37)(34,38);(1,3)(2,6)(4,8)(5,7)(9,14)(10,17)(11,20)(12,23)(13,18)(15,24)(16,19)(21,22)(25,30,31,28)(26,33,34,32)(27,37,38,36)(29,40,35,39)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8666299);