#8743441

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

4577 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=gi,f^2=g,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, 8650637);