#8718346

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

4582 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)(3,7)(4,6)(5,8)(9,10,15,16)(11,14,12,17)(13,21,23,18)(19,22,20,24)(25,26,30,34)(27,29,33,31)(28,36,37,32)(35,38,40,39);(1,2,4,6)(3,5,8,7)(9,11)(10,14)(12,15)(13,19)(16,17)(18,24)(20,23)(21,22)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,39)(37,40);(1,2)(3,5)(4,6)(7,8)(10,16)(11,12)(18,21)(19,20)(28,37)(32,36)(35,40)(38,39);(1,3,4,8)(2,5,6,7)(9,12,15,11)(10,17,16,14)(13,20,23,19)(18,22,21,24)(25,28)(26,32)(27,35)(29,39)(30,37)(31,38)(33,40)(34,36);(9,13,15,23)(10,18,16,21)(11,19,12,20)(14,24,17,22)(25,27)(26,29)(28,35)(30,33)(31,34)(32,39)(36,38)(37,40);(9,14,15,17)(10,12,16,11)(13,22,23,24)(18,19,21,20)(25,29)(26,33)(27,34)(28,38)(30,31)(32,35)(36,40)(37,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,d^2=i,e^2=h,f^2=h,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, 8625542);