#9010578

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

4565 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)(3,8)(4,7)(5,6)(9,10,14,17)(11,15,18,22)(12,16,21,13)(19,23,24,20)(25,26,30,34)(27,33,37,31)(28,29,39,32)(35,40,38,36);(3,4)(7,8)(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24)(25,27,30,37)(26,31,34,33)(28,35,39,38)(29,36,32,40);(1,3)(2,7)(4,6)(5,8)(11,18)(12,21)(13,16)(15,22)(25,28,30,39)(26,32,34,29)(27,35,37,38)(31,40,33,36);(3,4)(7,8)(9,12,14,21)(10,16,17,13)(11,19,18,24)(15,23,22,20)(25,28)(26,32)(27,35)(29,34)(30,39)(31,40)(33,36)(37,38);(1,4,6,3)(2,7,5,8)(9,13,14,16)(10,12,17,21)(11,20,18,23)(15,19,22,24)(25,27)(26,33)(28,38)(29,36)(30,37)(31,34)(32,40)(35,39);(1,5,6,2)(3,7,4,8)(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24)(25,29,30,32)(26,28,34,39)(27,36,37,40)(31,35,33,38)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=g,d^2=ghi,e^2=i,f^2=h,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, 8917774);