#9028789

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

4619 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=i,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=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, 8935985);