#8183802

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

4573 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:

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

Polycyclic presentation:

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

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, 8090998);