#8806824

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

4313 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=ghi,d^2=hi,e^2=ghi,f^2=h,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, 8714020);