#8930509

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

4189 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8837705);