#9028372

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

4519 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)(11,18)(12,20)(13,17)(14,15)(16,21)(19,23)(22,24)(25,26)(27,32)(28,33)(29,38)(30,31)(34,40)(35,39)(36,37);(1,2,5,4)(3,7,8,6)(9,11,15,17)(10,13,14,18)(12,19,21,24)(16,23,20,22)(25,26)(27,32)(28,36)(29,34)(30,31)(33,37)(35,39)(38,40);(1,3,5,8)(2,6,4,7)(10,14)(12,21)(13,18)(19,24)(26,31)(27,35)(28,37)(29,40);(3,8)(6,7)(9,12,15,21)(10,16,14,20)(11,19,17,24)(13,23,18,22)(25,27,30,35)(26,32,31,39)(28,38,37,34)(29,36,40,33);(1,3)(2,7)(4,6)(5,8)(9,13,15,18)(10,17,14,11)(12,22,21,23)(16,19,20,24)(25,28)(26,33)(27,34)(29,39)(30,37)(31,36)(32,40)(35,38);(1,4,5,2)(3,6,8,7)(9,14,15,10)(11,18,17,13)(12,20,21,16)(19,22,24,23)(25,29,30,40)(26,34,31,38)(27,36,35,33)(28,39,37,32)

Polycyclic presentation:

2^9|b^2=hi,c^2=ghi,d^2=h,e^2=g,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, 8935568);