#8697469

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

4341 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gi,d^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8604665);