#8159943

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

4628 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:

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

Polycyclic presentation:

2^9|b^2=gh,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8067139);