#8898667

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

4606 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)(3,8)(4,7)(5,6)(10,11)(15,17)(16,18)(22,23)(25,26,30,32)(27,31,36,35)(28,33,39,38)(29,34,40,37);(1,2,6,5)(3,7,4,8)(9,10,14,11)(12,15,20,17)(13,16,21,18)(19,22,24,23)(25,27)(26,31)(28,37)(29,33)(30,36)(32,35)(34,39)(38,40);(1,3,6,4)(2,7,5,8)(13,21)(16,18)(19,24)(22,23)(26,32)(27,36)(33,38)(34,37);(1,2,6,5)(3,7,4,8)(9,11,14,10)(12,15,20,17)(13,18,21,16)(19,22,24,23)(25,28)(26,33)(27,37)(29,31)(30,39)(32,38)(34,36)(35,40);(1,4,6,3)(2,7,5,8)(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,23)(21,24)(25,29,30,40)(26,34,32,37)(27,38,36,33)(28,35,39,31);(1,5,6,2)(3,7,4,8)(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,27)(26,35)(28,37)(29,38)(30,36)(31,32)(33,40)(34,39)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=hi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8805863);