#8183821

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

4523 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(2,4)(6,8)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,23)(21,22)(25,26,31,33)(27,34,28,30)(29,37,39,32)(35,38,36,40);(3,7)(6,8)(9,11,15,12)(10,16,17,18)(13,20,24,22)(14,21,19,23)(25,26,31,33)(27,34,28,30)(29,32,39,37)(35,40,36,38);(1,2,5,4)(3,6,7,8)(10,17)(14,19)(16,18)(21,23)(25,27,31,28)(26,30,33,34)(29,35,39,36)(32,38,37,40);(1,3,5,7)(2,6,4,8)(9,12,15,11)(10,18,17,16)(13,22,24,20)(14,23,19,21)(25,28,31,27)(26,30,33,34)(29,36,39,35)(32,38,37,40);(1,4,5,2)(3,8,7,6)(9,13)(10,19)(11,20)(12,22)(14,17)(15,24)(16,23)(18,21)(25,29,31,39)(26,32,33,37)(27,35,28,36)(30,38,34,40);(9,14,15,19)(10,13,17,24)(11,21,12,23)(16,20,18,22)(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=gi,c^2=gh,d^2=h,e^2=gh,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, 8091017);