#8195381

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

4506 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)(3,4)(5,6)(7,8)(9,10,14,18)(11,16,12,15)(13,17,21,24)(19,23,20,22)(25,26,29,30)(27,31,34,37)(28,32,36,38)(33,39,35,40);(2,6)(3,7)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(26,30)(31,37)(32,38)(39,40);(1,3,5,7)(2,4,6,8)(9,11,14,12)(10,15,18,16)(13,20,21,19)(17,23,24,22)(25,27,29,34)(26,31,30,37)(28,35,36,33)(32,40,38,39);(1,4)(2,7)(3,6)(5,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,26)(27,31)(28,32)(29,30)(33,39)(34,37)(35,40)(36,38);(9,12,14,11)(10,16,18,15)(13,20,21,19)(17,23,24,22);(9,13,14,21)(10,17,18,24)(11,19,12,20)(15,22,16,23)(25,28,29,36)(26,32,30,38)(27,33,34,35)(31,39,37,40)

Polycyclic presentation:

2^9|a^2=gi,c^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8102577);