#8709102

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

4645 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^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, 8616298);