#8873644

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4326 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=ghi,d^2=hi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8780840);