#9057878

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) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=hi,d^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8965074);