#8709118

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

4485 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)(11,18)(12,20)(13,17)(14,15)(16,22)(19,23)(21,24)(25,26)(27,28)(29,39)(30,33)(31,34)(32,38)(35,40)(36,37);(1,2,4,5)(3,6,7,8)(10,15)(11,17)(16,20)(19,24)(25,27,31,32)(26,28,34,38)(29,35,33,37)(30,36,39,40);(1,2)(3,6)(4,5)(7,8)(10,15)(11,17)(16,20)(19,24)(25,28,31,38)(26,32,34,27)(29,36,33,40)(30,37,39,35);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,19)(14,17)(15,18)(16,23)(20,21)(22,24)(25,28)(26,27)(29,36)(30,35)(31,38)(32,34)(33,40)(37,39);(1,3,4,7)(2,6,5,8)(9,12,14,22)(10,16,15,20)(11,19,17,24)(13,23,18,21)(25,29)(26,30)(27,35)(28,36)(31,33)(32,37)(34,39)(38,40);(9,13)(10,17)(11,15)(12,21)(14,18)(16,19)(20,24)(22,23)(25,30)(26,33)(27,36)(28,37)(29,34)(31,39)(32,40)(35,38)

Polycyclic presentation:

2^9|b^2=i,c^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8616314);