#8850856

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

4507 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)(9,10,14,18)(11,12,16,15)(13,21,23,17)(19,22,24,20)(25,26,30,33)(27,31,35,28)(29,32,38,40)(34,39,37,36);(2,5)(3,8)(9,11,14,16)(10,15,18,12)(13,19,23,24)(17,22,21,20)(25,27)(26,31)(28,33)(29,37)(30,35)(32,36)(34,38)(39,40);(1,3,6,8)(2,7,5,4)(9,12,14,15)(10,16,18,11)(13,20,23,22)(17,24,21,19)(25,26,30,33)(27,28,35,31)(29,32,38,40)(34,36,37,39);(1,4)(2,3)(5,8)(6,7)(9,12)(10,11)(13,22)(14,15)(16,18)(17,24)(19,21)(20,23)(25,28,30,31)(26,27,33,35)(29,36,38,39)(32,34,40,37);(1,5,6,2)(3,4,8,7)(9,13,14,23)(10,17,18,21)(11,19,16,24)(12,20,15,22)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,38)(39,40);(1,4)(2,3)(5,8)(6,7)(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,22)(23,24)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,40)(35,37)

Polycyclic presentation:

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

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, 8758052);