#7651839

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

4396 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=g,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7559035);