#8930320

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

4507 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,d^2=ghi,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, 8837516);