#8934545

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

4570 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8841741);