#9515616

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	#52387	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4058 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,e^2=ghi,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, 9422812);