#8999399

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

4437 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=g,e^2=hi,f^2=gh,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, 8906595);