#9779357

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

4160 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,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, 9686553);