#8739341

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

4441 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=gh,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8646537);