#8739389

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

4314 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)(3,7)(9,10,14,15)(11,13,17,23)(12,19,21,16)(18,20,24,22)(25,26)(27,32)(28,34)(29,33)(30,35)(31,36)(37,38)(39,40);(1,2,4,5)(3,6,7,8)(10,15)(11,17)(16,19)(18,24)(25,27,31,29)(26,32,36,33)(28,39,35,37)(30,38,34,40);(1,2)(3,6)(4,5)(7,8)(10,15)(11,17)(16,19)(18,24)(25,27,31,29)(26,33,36,32)(28,37,35,39)(30,38,34,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,18)(14,17)(15,23)(16,22)(19,20)(21,24)(25,28,31,35)(26,34,36,30)(27,37,29,39)(32,38,33,40);(1,3,4,7)(2,6,5,8)(9,12,14,21)(10,16,15,19)(11,18,17,24)(13,22,23,20)(25,29,31,27)(26,32,36,33)(28,39,35,37)(30,40,34,38);(9,13)(10,17)(11,15)(12,20)(14,23)(16,18)(19,24)(21,22)(25,30,31,34)(26,35,36,28)(27,38,29,40)(32,37,33,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,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, 8646585);