#8273508

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

4451 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8180704);