#8772804

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

4417 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8680000);