#8898335

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

4727 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8805531);