#8908390

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

4480 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)(11,16)(12,22)(13,17)(14,18)(15,21)(19,20)(23,24)(25,26,29,33)(27,30,36,39)(28,32,37,31)(34,40,35,38);(1,2,4,6)(3,5,8,7)(9,11)(10,15)(12,19)(13,20)(14,21)(16,18)(17,23)(22,24)(25,27,29,36)(26,30,33,39)(28,35,37,34)(31,40,32,38);(1,2)(3,5)(4,6)(7,8)(12,17)(13,22)(19,23)(20,24)(25,28,29,37)(26,31,33,32)(27,34,36,35)(30,38,39,40);(1,3,4,8)(2,5,6,7)(9,12,14,17)(10,13,18,22)(11,19,21,23)(15,20,16,24)(25,28)(26,32)(27,35)(29,37)(30,38)(31,33)(34,36)(39,40);(9,11)(10,16)(12,23)(13,20)(14,21)(15,18)(17,19)(22,24)(25,27)(26,30)(28,34)(29,36)(31,38)(32,40)(33,39)(35,37);(9,13)(10,17)(11,20)(12,18)(14,22)(15,23)(16,19)(21,24)(25,26)(27,30)(28,32)(29,33)(31,37)(34,40)(35,38)(36,39)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8815586);