#9028071

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

4421 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)(11,17)(12,16)(13,14)(15,18)(19,24)(20,23)(21,22)(25,26,29,33)(27,32,35,30)(28,36,38,31)(34,39,37,40);(1,2,5,4)(3,7,8,6)(11,18)(15,17)(19,20)(23,24)(25,26)(27,30)(28,31)(29,33)(32,35)(34,39)(36,38)(37,40);(1,3,5,8)(2,6,4,7)(9,11,14,18)(10,15,13,17)(12,20,22,19)(16,24,21,23)(25,27,29,35)(26,30,33,32)(28,37,38,34)(31,40,36,39);(3,8)(6,7)(9,12,14,22)(10,16,13,21)(11,19,18,20)(15,23,17,24)(25,28,29,38)(26,31,33,36)(27,34,35,37)(30,39,32,40);(1,3)(2,7)(4,6)(5,8)(9,11)(10,17)(12,20)(13,15)(14,18)(16,23)(19,22)(21,24)(25,27)(26,32)(28,37)(29,35)(30,33)(31,39)(34,38)(36,40);(1,4,5,2)(3,6,8,7)(9,13,14,10)(11,15,18,17)(12,21,22,16)(19,23,20,24)(25,26)(27,32)(28,31)(29,33)(30,35)(34,40)(36,38)(37,39)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gi,d^2=h,f^2=i,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, 8935267);