#8898289

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

4411 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gi,d^2=gh,e^2=h,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, 8805485);