#8943133

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

4375 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gi,e^2=h,f^2=g,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, 8850329);