#8818090

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

4361 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=i,d^2=gi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8725286);