#8824170

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

4418 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=hi,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8731366);