#9515320

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
prime radical	#45768	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4061 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)(11,17)(12,22)(13,15)(14,19)(16,20)(18,23)(21,24)(25,26)(27,35)(28,32)(29,33)(30,34)(31,38)(36,40)(37,39);(1,2)(3,6)(4,5)(7,8)(9,11,16,13)(10,17,20,15)(12,14,23,21)(18,24,22,19)(25,26,30,34)(27,31,38,35)(28,32,29,33)(36,39,37,40);(2,5)(3,7)(9,12,16,23)(10,18,20,22)(11,21,13,14)(15,24,17,19)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(3,7)(6,8)(9,13,16,11)(10,15,20,17)(12,21,23,14)(18,19,22,24)(25,27)(26,31)(28,36)(29,37)(30,38)(32,39)(33,40)(34,35);(1,3,4,7)(2,6,5,8)(9,14,16,21)(10,19,20,24)(11,12,13,23)(15,18,17,22)(25,28,30,29)(26,32,34,33)(27,36,38,37)(31,39,35,40);(1,2)(3,6)(4,5)(7,8)(9,15)(10,11)(12,19)(13,20)(14,18)(16,17)(21,22)(23,24)(25,29,30,28)(26,33,34,32)(27,37,38,36)(31,40,35,39)

Polycyclic presentation:

2^9|b^2=gh,c^2=h,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,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, 9422516);