#8384230

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

4495 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8291426);