#8119329

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

4502 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=hi,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8026525);