#8942660

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

4468 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)(3,7)(4,6)(5,8)(9,10,13,16)(11,14,17,18)(12,15,20,22)(19,21,23,24);(1,2,4,6)(3,5,8,7)(9,10)(11,14)(12,15)(13,16)(17,18)(19,21)(20,22)(23,24)(25,26,30,31)(27,28,32,35)(29,33,37,38)(34,36,39,40);(1,2,4,6)(3,7,8,5)(11,17)(12,20)(14,18)(15,22)(26,31)(28,35)(29,37)(34,39);(1,2,4,6)(3,5,8,7)(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,22)(21,23)(25,27)(26,28)(29,34)(30,32)(31,35)(33,36)(37,39)(38,40);(1,2)(3,5)(4,6)(7,8)(9,11,13,17)(10,14,16,18)(12,19,20,23)(15,21,22,24)(25,28)(26,32)(27,31)(29,36)(30,35)(33,39)(34,38)(37,40);(1,3)(2,5)(4,8)(6,7)(9,12,13,20)(10,15,16,22)(11,19,17,23)(14,21,18,24)(25,29)(26,33)(27,34)(28,36)(30,37)(31,38)(32,39)(35,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=h,d^2=h,e^2=gh,f^2=gh,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, 8849856);