#8799448

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

4362 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=ghi,e^2=gi,f^2=h,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, 8706644);