#8905219

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

4648 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=gh,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8812415);