#9945541

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

4470 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)(6,7)(9,10)(11,20)(12,17)(13,18)(14,16)(15,19)(21,22)(23,24)(26,30)(28,35)(31,37)(33,36);(1,2,5,4)(3,7,8,6)(9,11,15,14)(10,16,19,20)(12,23,18,21)(13,24,17,22)(26,30)(31,37)(32,38)(39,40);(1,2,5,4)(3,7,8,6)(25,26,29,30)(27,31,34,37)(28,32,35,38)(33,39,36,40);(1,3)(2,6)(4,7)(5,8)(9,12,15,18)(10,17,19,13)(11,21,14,23)(16,24,20,22)(25,26)(27,31)(28,32)(29,30)(33,39)(34,37)(35,38)(36,40);(1,3)(2,7)(4,6)(5,8)(9,13,15,17)(10,18,19,12)(11,22,14,24)(16,23,20,21)(25,27,29,34)(26,31,30,37)(28,36,35,33)(32,40,38,39);(1,4,5,2)(3,7,8,6)(9,14,15,11)(10,16,19,20)(12,23,18,21)(13,22,17,24)(25,28)(26,32)(27,33)(29,35)(30,38)(31,39)(34,36)(37,40)

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 9852737);