#9712834

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	#51585	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4168 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)(3,8)(4,6)(5,7)(9,10,16,20)(11,17,23,24)(12,13,14,18)(15,22,21,19)(25,26,32,35)(27,36,29,30)(28,37,38,33)(31,39,40,34);(1,3,5,6)(2,4,7,8)(9,11)(10,17)(12,22)(13,21)(14,19)(15,18)(16,23)(20,24)(25,27,32,29)(26,30,35,36)(28,34,38,39)(31,37,40,33);(9,12,16,14)(10,18,20,13)(11,19,23,22)(15,17,21,24)(25,28,32,38)(26,33,35,37)(27,34,29,39)(30,31,36,40);(9,13,16,18)(10,12,20,14)(11,21,23,15)(17,22,24,19)(25,29,32,27)(26,30,35,36)(28,34,38,39)(31,33,40,37);(1,2)(3,4)(5,7)(6,8)(9,14,16,12)(10,18,20,13)(11,22,23,19)(15,17,21,24)(25,30)(26,27)(28,31)(29,35)(32,36)(33,34)(37,39)(38,40);(1,4)(2,6)(3,7)(5,8)(9,15,16,21)(10,19,20,22)(11,18,23,13)(12,17,14,24)(25,31)(26,34)(27,37)(28,36)(29,33)(30,38)(32,40)(35,39)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9620030);