#9717416

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

4228 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=g,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9624612);