#8919482

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

4530 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)(9,10,15,19)(11,20,13,16)(12,14,18,17)(21,24,23,22);(1,3,6,5)(2,7,4,8)(9,11,15,13)(10,16,19,20)(12,21,18,23)(14,22,17,24)(25,26,30,31)(27,32,34,39)(28,29,33,38)(35,40,37,36);(1,3,6,5)(2,8,4,7)(12,18)(14,17)(21,23)(22,24)(26,31)(27,34)(29,38)(35,37);(9,12,15,18)(10,17,19,14)(11,21,13,23)(16,24,20,22)(25,27)(26,32)(28,37)(29,36)(30,34)(31,39)(33,35)(38,40);(1,4,6,2)(3,7,5,8)(9,13,15,11)(10,16,19,20)(12,21,18,23)(14,24,17,22)(25,28,30,33)(26,29,31,38)(27,35,34,37)(32,40,39,36);(1,5,6,3)(2,7,4,8)(9,14,15,17)(10,18,19,12)(11,22,13,24)(16,23,20,21)(25,29)(26,33)(27,36)(28,31)(30,38)(32,35)(34,40)(37,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,d^2=g,e^2=ghi,f^2=gh,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, 8826678);