#9057900

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

4561 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)(11,16)(12,22)(13,18)(14,15)(17,23)(19,21)(20,24)(25,26,30,33)(27,28,31,34)(29,32,38,37)(35,36,39,40);(1,3,5,8)(2,6,7,4)(9,11)(10,16)(12,20)(13,17)(14,21)(15,19)(18,23)(22,24)(25,27)(26,28)(29,35)(30,31)(32,36)(33,34)(37,40)(38,39);(1,2)(3,4)(5,7)(6,8)(10,14)(11,19)(12,23)(13,24)(25,26,30,33)(27,34,31,28)(29,37,38,32)(35,36,39,40);(1,4)(2,3)(5,6)(7,8)(9,12,15,23)(10,17,14,22)(11,20,19,18)(13,21,24,16)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,38)(39,40);(1,3)(2,4)(5,8)(6,7)(9,13,15,24)(10,18,14,20)(11,17,19,22)(12,16,23,21)(25,28)(26,31)(27,33)(29,36)(30,34)(32,39)(35,37)(38,40);(1,4)(2,3)(5,6)(7,8)(9,14,15,10)(11,21,19,16)(12,22,23,17)(13,20,24,18)(25,29,30,38)(26,32,33,37)(27,35,31,39)(28,36,34,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=hi,d^2=gh,e^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, 8965096);