#9057804

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

4764 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)(11,15)(12,19)(13,21)(14,16)(17,22)(18,24)(20,23)(25,26,30,34)(27,33,37,31)(28,32,39,29)(35,36,38,40);(1,3,5,8)(2,6,7,4)(9,11,14,12)(10,15,16,19)(13,18,22,20)(17,23,21,24)(25,27,30,37)(26,31,34,33)(28,35,39,38)(29,36,32,40);(1,3)(2,4)(5,8)(6,7)(10,16)(11,12)(13,22)(23,24)(25,28,30,39)(26,32,34,29)(27,35,37,38)(31,40,33,36);(1,4)(2,3)(5,6)(7,8)(9,12,14,11)(10,15,16,19)(13,20,22,18)(17,23,21,24)(25,28)(26,32)(27,35)(29,34)(30,39)(31,40)(33,36)(37,38);(9,10)(11,15)(12,19)(13,17)(14,16)(18,23)(20,24)(21,22)(25,27)(26,33)(28,38)(29,36)(30,37)(31,34)(32,40)(35,39);(9,13,14,22)(10,17,16,21)(11,18,12,20)(15,23,19,24)(25,29,30,32)(26,28,34,39)(27,36,37,40)(31,35,33,38)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=ghi,d^2=h,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, 8965000);