#8402504

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

4403 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:

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

Polycyclic presentation:

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

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, 8309700);