#8734917

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

4283 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=h,e^2=h,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, 8642113);